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+{"text":"\\section{\\label{sec:level1}First-level heading}\n\nSilicon Carbide has become the wide band gap (WBG) semiconductor with the most mature technology \\cite{Kimoto2015MaterialAnnealing} and it is finally ready to penetrate the power devices market after more than two decades elapsed as faint promise of next generation power electronics \\cite{Chelnokov1997OverviewElectronics,Willander2006SiliconApplications}. Indeed, it is expected that SiC will reach about 10\\% of the Si market by 2025 with a compound annual growth rate (CAGR) of about 40\\% from 2020 to 2022 \\cite{Bhalla2018StatusTechnology}. But, to continue the development of SiC technology and sustain the improvements in efficiency and performance of WBG based devices, research efforts need to be continued even at the level of material physical understanding. In fact, the material maturity process has been quite slow. An evident reason is the intrinsic complexity of this semiconductor compound: it occurs only rarely in nature and it has more than 200 polytypes \\cite{N.W.JeppsandT.F.Page1983NoTitle}. If properly understood and controlled, polytypism actually provides an added value. In fact, the most common SiC polytypes (3C-, 4H- and 6H-SiC) cover a range of band-gap from about 2.3 to 3.2 eV and thus they are suitable both for low and high-power devices.\\\\\nBesides the scientific interest, investigating SiC polytypism and understanding its driving force is crucial to correctly predict the energetics of extended defects in SiC, particularly stacking faults (SFs), which are a main concern of this WBG semiconductor since they cause deterioration and eventually failure of the devices after relative long operational time  \\cite{Ishida2002InvestigationEpilayers,Nagasawa2008FabricationDefects,Eriksson2011Electrical3C-SiC}.\\\\\nSiC polytypes consist of identical double layers with different stacking sequences, thus generating orders of SiC tetrahedrons with different orientation, as highlighted in blue and red color in Fig.\\ref{sym}. Truly, SFs are wrong sequences of the double layers or in other words, they can be seen as inclusions of few layers of a SiC polytype in the perfect layer stacking of another polytype (see inset in Fig.\\ref{sym}).  Due to the small-scale energy difference between the stacking sequences of double layers and hence between the different SiC polytypes, as discussed below, perturbations of the ideal stacking sequence during SiC crystal growth are very likely. This is another reason why SFs are so critical in this material. \\\\\n\\begin{figure}[b]\n  \\centering\n    \\includegraphics[width=0.48\\textwidth]{SiCsymmDrawing4.png}\n\\caption{\\label{sym} Tetrahedral stacking sequences of 3C-, 2H-, 4H- and 6H-SiC. The red and blue triangles highlight the twinned or normal tetrahedra and correspond to down or up spin configurations of the SiC layers, according to the axial next-nearest neighbor Ising (ANNNI) model \\cite{Fisher1980InfinitelyModei}\nThe inset shows the stacking sequence of 3C-SiC polytype including an intrinsic, extrinsic and double extrinsic stacking fault (labeled ISF, ESF and ESFd respectively). \n}\n\\end{figure}\nThe literature on the thermodynamic stability and polytypism of SiC is abundant, encouraged by the physical and technological interests evidenced above. Still, the predictions on the free energy of SiC polytypes are misleading and the energetic hierarchy reported by different theoretical methods is often inconsistent with some experimental observations. In fact, a paradox concerning SiC polytypes \\cite{Heine1991NoTitle,Cheng1990AtomicPolytypes, Fan2014SiliconNanostructures,Zywietz1996InfluenceCarbide} has been often discussed, dealing with the theoretical predictions of hexagonal (6H) SiC as the most stable polytype and of the cubic (3C) one as not stable at any temperature \\cite{Cheng1990AtomicPolytypes,Bechstedt1997PolytypismCarbide,Feng2004SiCApplications}. Contrary, experiments have shown that the cubic (3C) structure does grow in preference to all others and only at very high temperatures hexagonal phases, i.e. 4H and 6H polytypes, have been observed to prevail \\cite{Boulle2010QuantitativeCrystals,Kanaya1991ControlledDiffraction,Yakimova2000PolytypeBoules,Kado2013High-SpeedMelt,Kusunoki2014Top-seededTechnique,Heine1991NoTitle,Bechstedt1997PolytypismCarbide}. Thereby,  SiC polytype stability at different temperatures remains unclear. Different arguments has been proposed over the years to explain polytypism and polytypic transformation in SiC \\cite{Bechstedt1997PolytypismCarbide,Cheng1988Inter-layerPolytypes,Heine1991NoTitle,Ching2006TheSiC,Zywietz1996InfluenceCarbide,Bernstein2005Tight-bindingCarbide,Lindefelt2003StackingPo}, including  the motion of partial dislocations \\cite{Pirouz1993PolytypicTEM,Pirouz1997PolytypicSiC,Boulle2010QuantitativeCrystals,Boulle2013PolytypicSimulations} and impurity effects \\cite{Heine1991NoTitle} on crystal growth. Nonetheless, inconsistencies between theory and experiments still remain. \n\nRecently it was shown that density functional theory (DFT) calculations \\cite{SM} including the van der Waals (vdW) correction do predict the 3C phase to have the lowest free energy at T=0K \\cite{Kawanishi2016EffectPolytypes}. This intriguing result points out the importance of considering long-range interactions when comparing different SiC polytypes.\n\nIn this Letter we show that, by consistently adding the entropic contribution to the free energy, within a DFT approach that includes vdW corrections, the full T-dependent hierarchy of polytypes is correctly predicted, showing a cross-over between 3C and 6H (or 4H) phases at typical experimental temperatures. As detailed below, the present results allow for better understanding of the physics behind SiC polytypism by simple thermodynamics considerations, highlighting the correlation between hexagonality, vibrational properties and cohesive energy.\nMoreover, calculations of the stability of SiC SFs reveal a T-dependent behavior, intimately correlated to the polytypic stability, and they provide insight into optimal growth temperatures for lowering the density of such defects.      \n\n\n\n\n\nIn Table \\ref{tab:table1} the lattice constants of 3C-, 6H-, 4H- and 2H-SiC are reported, both calculated at the generalized gradient approximation (GGA) level \\cite{Perdew1996GeneralizedSimple} and with the semiempirical Grimme's method, which employs GGA-type density functional constructed with a long-range dispersion correction \\cite{Barone2009RoleCases,Grimme2006SemiempiricalCorrection} and accounting for vdW interactions.\n\\begin{table}[!t\n\\caption{\\label{tab:table1}%\nLattice constants (\\text{\\AA}), total energy $\\Delta E_T$ (relative to 3C-SiC), hexagonality and heat of formation $\\Delta H_f$ (meV\/SiC) of 3C-, 6H-, 4H- and 2H-SiC.}\n\\begin{ruledtabular}\n\\begin{tabular}{ l  c c c}\n\\textrm{Polytype   \\hspace{15pt} hex} & \n\\textrm{a,c} &\n\\textrm{$\\Delta E_T$}&\n\\textrm{$\\Delta H_f$} \\\\\n\n\\colrule\n\\textbf{3C-SiC} \\hspace{18pt} 0\\%\\\\\nGGA & 4.377 & 0 & -402\\\\\n GGA(vdW) & 4.352 & 0 & -785\\\\\n exp\\cite{o1960silicon, Kleykamp1998GibbsModifications,Greenberg1970TheCalorimetry,2014CRCData}  & 4.3596  &  & -650, -758, -771\\\\\n\\colrule\n\n\\textbf{6H-SiC} \\hspace{18pt} 33\\%\\\\ \nGGA & 3.093,15.178 & -1.7 & -404 \\\\\n GGA(vdW) & 3.075,15.105 & 1.4 & -784\\\\\n exp\\cite{o1960silicon, Kleykamp1998GibbsModifications,Greenberg1970TheCalorimetry,Tairov1983ProgressCrystals} & 3.080,15.117 & & -676, -747, -771 \\\\\n\\colrule\n\n\\textbf{4H-SiC} \\hspace{18pt} 50\\%\n\\\\GGA & 3.092,10.123 & -1.8 & -404\\\\\n  GGA(vdW) & 3.074,10.079 & 2.9  & -783\\\\\n exp\\cite{Zemann1965iCrystalWyckoff,Tairov1983ProgressCrystals,2014CRCData} & 3.073,10.053 & & -650, -689\\\\\n\\colrule\n\n\\textbf{2H-SiC} \\hspace{18pt} 100\\%\\\\\n GGA & 3.090,5.072 & 5.8 & -396\\\\\n  GGA(vdW) & 3.072,5.056 & 15.1 & -770\\\\\n exp\\cite{Schulz1979STRUCTUREZnO} & 3.079,5.053 \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nIt is evident that the agreement between theoretical and experimental lattice parameters improves considerably for vdW-corrected DFT simulations. These improvements are even more appreciable by looking at the heat of formation ($\\Delta H_f$) in Table \\ref{tab:table1}, which is severely underestimated by GGA but in very good agreement with experiments in the case of vdW-corrected simulations.\nNote that, not only the magnitude of the heat of formation but also the order of the values calculated for the different polytypes changes whether or not the simulations include the vdW correction. This is very clear looking at the total energy of the different polytypes ($\\Delta E_T$) in Table \\ref{tab:table1}, calculated as a relative value with respect to the total energy of 3C-SiC. While $\\Delta E_T$ values calculated by GGA are all negative except for 2H-SiC, thus predicting 3C-SiC as the least stable polytype after the 2H-SiC, the vdW-corrected simulations give all positive $\\Delta E_T$. Hence, 3C-SiC turns out to be the most stable polytype. This is well in agreement both with experimental evidences inferring 3C-SiC as the most stable SiC structure in the nuclear stage \\cite{Tairov1983ProgressCrystals} and with a recent theoretical work \\cite{Kawanishi2016EffectPolytypes}.\nNevertheless, the predicted energetic hierarchy at T=0K is still not sufficient to understand the competition in stability of SiC polytypes at higher temperature, as probed experimentally \\cite{Boulle2010QuantitativeCrystals,Boulle2013PolytypicSimulations,Kanaya1991ControlledDiffraction,Yakimova2000PolytypeBoules,Kado2013High-SpeedMelt,Kusunoki2014Top-seededTechnique}. Thus, the variation of the entropic contributions with temperature for the different polytypes becomes crucial. This has been included in Fig.\\ref{HelmohotzF}, where the Helmholtz free energy for the SiC polytypes is plotted as a difference between the values of the 2H, 4H and 6H polytypes and that of 3C-SiC. \\\\\nThe typical expression of the Helmholtz free energy $F(T)=U-TS$, with $U$ the internal energy and $TS$ the product of temperature and entropy, can be also reformulated as $F(T)=U_0+U_{vib}-TS$, where the internal energy $U$ is split into the vibrational internal energy ($U_{vib}$) and static internal energy ($U_0$), with the latter corresponding to the total DFT energy of the SiC polytypes at their GGA-vdW equilibrium geometry. Also the other terms of the Helmholtz free energy can be conveniently calculated by DFT \\cite{Baroni2009ThermalPhonons}. In fact, one can additionally formulate $F(T)$ as a sum of the electronic and vibrational contributions, thus $F(T)=F_{el}(T)+F_{vib}(T)$. The former term $F_{el}(T)$ can be reasonably approximated by its zero-temperature limit $(U_0)$ \\cite{Bechstedt1997PolytypismCarbide}, by neglecting the electronic entropy; the vibrational contribution, which then correspond to $F_{vib}(T)=U_{vib}-TS$, can be calculated by the quasi-harmonic approximation as \\cite{Born1955DynamicalLattices,Feng2004SiCApplications}:\n\\begin{equation}\n\\nonumber \nF_{vib}(T)=\\int _{0}^{\\infty}g(\\omega)  \\left [  \\frac{\\hbar\\omega}{2 K_B T} + \\ln \\left ( 1-e^{  \\frac{-\\hbar\\omega}{K_B T } } \\right ) \\right ] d\\omega \\thinspace;\n\\end{equation}\nwhere $\\omega$ is the phonon frequency and $g(\\omega)$ is the phonon density of states. In our simulations, $\\omega$ and $g(\\omega)$ are calculated in the framework of the density functional perturbation theory (DFPT) \\cite{Baroni1987Greens-functionSolids,Baroni2009ThermalPhonons}.\\\\\n\\begin{figure}[!t]\n  \\centering\n    \\includegraphics[width=0.48\\textwidth]{HelmotzFinale.png}\n\\caption{\\label{HelmohotzF} Difference between the Helmholtz free energy of 3C-SiC with respect to the value of 6H, 4H and 2H polytypes.\n\\end{figure}\nThe values at T=0K of the three curves plotted in Fig.\\ref{HelmohotzF} reveal that the zero-point internal energy (ZPE), which is the main contribution to $F_{vib}$ in the low temperature range, only slightly affects the static internal energy (cf. $\\Delta E_T$ in Table \\ref{tab:table1}) and does not change the energetic hierarchy of the SiC polytypes. But for temperatures above 500K the vibrational contribution ($F_{vib}$) becomes considerable and the Helmohltz free energies of the hexagonal polytypes get closer to the cubic one. Particularly, at temperature of about 1750K the difference between the free energies $(F_{3C}-F_{6H})$ crosses the 0 energy line, meaning that 6H polytype becomes thermodynamically more stable than 3C. The energy crossing between 4H- and 3C-SiC is predicted a bit higher in temperature, at about 2400K. Contrary, 2H- never becomes more stable than 3C-SiC in the temperature range considered, albeit their free energies get closer at higher temperatures. The comparison between hexagonal polytypes reveals that 2H-SiC is the least thermodynamically stable structure: the difference of its free energy and that of 4H or 6H polytype marginally decreases with the temperature. Contrary, the free energies of 6H and 4H polytypes get very close at temperatures around 2500K. Our T-dependent diagram of the polytypic stability plotted in Fig.\\ref{HelmohotzF} is in excellent agreement with several experimental evidences such as the preferential growth at temperatures below $\\sim$1850K of the 3C polytype over all others \\cite{Heine1991NoTitle,Bechstedt1997PolytypismCarbide,Cheng1988Inter-layerPolytypes}, the higher stability of 4H and 6H polytypes at higher temperatures \\cite{Boulle2010QuantitativeCrystals,Boulle2013PolytypicSimulations,Kanaya1991ControlledDiffraction,Yakimova2000PolytypeBoules,Kado2013High-SpeedMelt,Kusunoki2014Top-seededTechnique}, and the rare appearance of 2H-SiC \\cite{Pirouz1997PolytypicSiC,Imade2009Liquid-phaseMelt}.  \n\nThe correlation between hexagonality and the observed trends in the Helmholtz free energy is elucidated by the calculated entropy, $S=-(\\delta F_{vib}\/\\delta T)$, reported in Fig.\\ref{FigEntropy}.\nAn opposite hierarchy of the entropy with respect to the cohesive energy at all temperatures is found. This is further supported by the general decreasing trend of phonon frequencies with hexagonality, which is evident in the phonon density of states (PDOS) plotted for the different polytypes in the region of the longitudinal optical (LO) branch \\cite{SM} in the inset of Fig.\\ref{FigEntropy}. The shift of the phonon frequencies is associated to a different strength of the interactions between hexagonally and cubically stacked layers, thus a correspondence between the lower hexagonality and the higher cohesion of the structure is evident.\nThe difference in the free energy at T=0 between 3C- and 2H-SiC is so large compared to their entropy difference that 2H remains less stable even at high temperature. On the contrary the much smaller difference between the static energy  of 3C and 6H (or 4H) is overcompensated by the larger entropy contribution of the latter.\nThis provides an intuitive picture to understand SiC polytypism: cubic SiC polytype have higher cohesive energy, higher stiffness and lower entropy; contrary, hexagonal (2H) polytype has the lowest cohesive energy, lower stiffness but a higher entropy; in between, the trends of the other two hexagonal polytypes investigated follow their percentage of hexagonality, with their higher entropy, as compared to 3C-SiC, leading to changes of the energetic hierarchy with the temperature.\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[width=0.48\\textwidth]{Entropy.png}\n\\caption{\\label{FigEntropy} Difference between the entropy of \\emph{x}H and 3C politypes. The inset shows the PDOS in the region near the LO band.}\n\\end{figure}\n\nThe correct prediction of the free energy of polytypes is a compelling need for the investigation of other essential aspects of SiC and it will be exploited below for studying the SFs stability. \nThe energetic cost of any error in the stacking sequence can be estimated by two different approaches: modeling a perturbed layer stacking by supercell structures, such as those illustrated in the inset of Fig.\\ref{sym}, and then calculating the total energy of the faulted supercell; alternatively one can calculate the SF energy according to the axial next-nearest neighbor Ising (ANNNI) model \\cite{Fisher1980InfinitelyModei}. In fact, the energy of the different polytypes, both pristine or faulted, can be expressed in terms of the interactions between SiC double layers. Accordingly, for the \\emph{x}H- (or \\emph{x}C-) SiC polytypes the total free energy is:\n\\begin{equation}\nE = E_0 - \\frac{1}{x}\\displaystyle\\sum_{i=1}^n \\displaystyle\\sum_{j=1}^{\\infty}J_j \\sigma_i \\sigma_{i+j} \\thinspace;\n\\label{eqn:Name}\n\\end{equation}\nwhere $E_0$ is a common reference energy and $J_j$ are the interaction energies between \\emph{i}th-neighbour double layers. The double layers are represented by a pseudospin $\\sigma_i$, which can be spin-up or spin-down (with value +1 or -1, respectively) according to the tetrahedron orientation that the layers form: $\\sigma_i=+1$ corresponds to a normal orientation (blue color in fig.\\ref{sym}) while $\\sigma_i=-1$ to a twinned one (red in fig.\\ref{sym}) \\cite{Cheng1988Inter-layerPolytypes, Pirouz1993PolytypicTEM}. For instance, 4H-SiC is represented by two spin-up and two spin-down, 6H- by three up and three down. In Eq.\\ref{eqn:Name}, spin coupling higher than third-order are usually neglected. The static free energies of Table \\ref{tab:table1} can be then used to obtain the $J_j$ values from Eq.\\ref{eqn:Name} and they allows one to calculate the free energies of faulted structures. Finally, stacking fault energies are estimated as the energy difference between the faulted and the pristine structure \\cite{Hong2000StackingCrystals,Boulle2013PolytypicSimulations,Lindefelt2003StackingPo} and they are listed in Table \\ref{tab:table2}. \n\\begin{table}[!t\n\\caption{\\label{tab:table2}%\nFormation energy (mJ m$^{-2}$) of SFs in 3C, 6H, and 4H-SiC. Values calculated by the ANNNI model (and GGA calculations with\/out vdW correction) or obtained by the supercell approach (with vdW) are reported. Experimental values from \\cite{Ning1990ExperimentalCrystals,Hong2000StackingCrystals} and other theoretical values from \\cite{Umeno2012Ab3C-SiC, Kackell1998StackingStudy,Lindefelt2003StackingPo} are also listed.}\n\\begin{ruledtabular}\n\\begin{tabular}{l|ccc|c|c}\n        & \\multicolumn{3}{c|}{3C-SiC} & 6H-SiC & 4H-SiC \\\\\n        & ISF     & ESF    & ESFd    & ISF    & ISF    \\\\ \\hline\nGGA     & 4.35    & -16.7 & -19.5    & 2.84    & 18.23        \\\\\nGGA vdW & 40.70  & 19.6  & 16.85   &    2.77    &   18.35    \\\\\nSuperc.& 40.21 & 19.62 & 17.04 & & \\\\\nExp.    & 34 &  &  &2.9$\\pm$0.6&14.7$\\pm$2.5 \\\\\nCalc.\\cite{Lindefelt2003StackingPo}  & -6.27 &   &   & 3.14 & 18.3\\\\\nCalc.\\cite{Umeno2012Ab3C-SiC}  & 10.3 & -7.83 & -11.6 &  &  \\\\\nCalc.\\cite{Kackell1998StackingStudy}  & -3.4 & -28 &  &  &  \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nWe also checked the reliability of the SF energies calculated by the ANNNI model comparing them with the corresponding value obtained by simulating the defected supercells of 3C-SiC illustrated in the inset of Fig.\\ref{sym}. Interestingly, the calculated SF energy values are all positive and in excellent  agreement with experimental estimations \\cite{Ning1990ExperimentalCrystals,Hong2000StackingCrystals} if the vdW correction is included in the DFT simulations. Contrary, SF energies obtained by bare GGA-DFT are very different, particularly for 3C-SiC. In fact, the formation energy of intrinsic stacking faults (ISFs) in 3C-SiC calculated by GGA is much lower than the corresponding value obtained including the vdW correction. For extrinsic stacking faults, both single (ESF) and double (ESFd), the SF energies turn even into negative values. This is not surprising if one goes through the literature of SFs in 3C-SiC, in which very small or even negative theoretical values of the ESF energy in 3C-SiC are well-accepted (see Table \\ref{tab:table2}). Instead, these SF energies are doubtful if compared with experiments \\cite{Ning1990ExperimentalCrystals}. Typically, SF energy is experimentally estimated by comparing the measured width of the stacking fault between the two terminating partial dislocations \\cite{Ning1990ExperimentalCrystals,Hong2000StackingCrystals} and its expectation by means of the dislocation theory for anisotropic elastic media \\cite{Ning1990ExperimentalCrystals,Hong2000StackingCrystals,P.Hirth1982TheoryDislocations}. Accordingly, SFs with negative formation energies should not have finite width, thus in evident contradiction with experiments. \\\\%Note also that the discrepancy between bare GGA and vdW-corrected calculations is appreciable for the SF energies of 3C-SiC but not much for the hexagonal structure, confirming the higher effect of the long-range interactions on the energetics of the SiC structures with lower hexagonality \\cite{Kawanishi2016EffectPolytypes}.\nFinally, by exploiting the ANNNI model and the T-dependent free energies presented above, we plot in Fig.\\ref{defF} the SF formation energies for the 3C-, 6H-, and 4H-SiC as a function of the temperature. Different trends in temperature between hexagonal and cubic polytypes are found: while the formation energies of SF in 6H- and 4H-SiC slightly increase with the temperature, for 3C-SiC the SF energies decrease substantially with the temperature and particularly for the ESFs, they become negative at temperature above 1750K. \\\\\n\\begin{figure}[!t]\n  \\centering\n    \\includegraphics[width=0.48\\textwidth]{SFenergy2.png}\n\\caption{\\label{defF} Formation energy of ISF in 6H- and 4H-SiC, and of ISF, ESF and ESFd in 3C-SiC.}\n\\end{figure}\nIn conclusion, we have shown that DFT calculations including the vdW correction predict a T-dependent hierarchy of SiC \npolytypes in perfect agreement with the experimental results. 3C-SiC is predicted to have the highest cohesive energy but the lowest entropy. At high temperature, the higher entropic contribution  to the free energy of the hexagonal polytypes stabilizes their structures, with the 6H and 4H-SiC becoming thermodynamically more stable than 3C-SiC. These results demonstrate the key role of the thermodynamics in determining SiC polytypism and contribute to finally reconcile theory and experiments. They are also essential for understanding SF stability in SiC, yielding positive formation-energy values for both ESF and ISF in 3C-SiC that are at variance with previous theoretical results, but in accord with experimental evidences. Moreover, the  formation energy of 3C-SiC SFs is predicted to decrease substantially with temperature, becoming lower than that predicted for 6H-SiC and eventually negative. Importantly, this indicates that too high deposition temperatures should be avoided in order to decrease SF density in 3C-SiC.\n\n\\clearpage\n\n\n\n\n\n\n\n\\begin{acknowledgements}\nAuthors acknowledge EU for founding the CHALLENGE project (3C-SiC Hetero-epitaxiALLy grown on silicon compliancE substrates and 3C-SiC substrates for sustaiNable wide-band-Gap powEr devices) within the EU's H2020 framework programme for research and innovation under grant agreement n. 720827.\n\\end{acknowledgements}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\t\n\t\\subsection{Dilating sets in diverse geometric contexts}\n\t\\label{sec:intro}\n\tIt is an intriguing geometric problem to understand the long-term distribution properties of a diversified range of progressively dilating shapes, when the ambient space in which they live is folded according to a prescribed rule. Formally, the framework underlying such a question can be laid down as follows. Consider a compact connected Riemannian manifold $S$ and a Riemannian covering space $N$ with covering projection $\\pi\\colon N\\to S$. The manifold $N$ plays the role of the ambient space, while $S$ is to be interpreted as the manifold resulting after a certain folding procedure operated on $N$. A homothety on $N$ is a diffeomorphism $h\\colon N\\to N$ transforming the Riemannian metric on $N$ into a scalar multiple thereof; if the rescaling ratio is equal to $1$, it simply reduces to a Riemannian isometry. Let now $(h_t)_{t\\in \\R_{>0}}$ be a collection of homotheties on $N$ whose scaling factor tends to infinity as $t$ does, and fix a Borel probability measure $\\mu$ on $N$. The latter should be thought of as describing quantitatively the original shape, whose dilations we wish to examine. For instance, $\\mu$ might be the renormalized volume measure on a finite-volume Riemannian submanifold of $N$, e.g.~a rectifiable curve. We then let $\\mu_t$ be the push-forward of $\\mu$  under the homothety $h_t$ and denote by $m_t$ the projection of $\\mu_t$ onto $S$, for any $t>0$.\n\t\n\tA mathematical formulation of our problem consists in asking for the limit points, as $t$ goes to infinity, of the family of measures $(m_t)_{t>0}$ in the topology of weak$^*$ convergence of probability measures on $S$. Somewhat less pretentiously, it is already interesting to determine sufficient geometric conditions on the initial measure $\\mu$ ensuring that the $m_t$ equidistribute as $t$ grows larger, that is, that they converge in the aforementioned topology to the renormalized volume measure $\\vol_S$ on $S$; this circumstance amounts pictorially to the fact that the measures $m_t$ (and hence, in an intuitive sense, their supports) fill up the folded space $S$ in the most uniform way.\n\t\n\t\\subsubsection*{The Euclidean case}\n\t\n\tTo the best of the authors' knowledge, the question formulated in the previous paragraph was first asked by Dennis Sullivan (cf.~\\cite{Randol}) in the zero-curvature setting of Euclidean spaces and tori, that is, when $S=\\T^{d}=\\R^{d}\/\\Z^d$ and $N=\\R^{d}$ for some integer $d\\geq 1$, and for $\\mu$ being the volume measure on a compact lower-dimensional submanifold of $\\R^{n}$. In this case, the transformations $h_t$ are the standard linear homotheties $x\\mapsto tx,\\;x\\in \\R^{d}$. The problem was originally addressed by Randol in~\\cite{Randol}, both in an Euclidean and in a hyperbolic setup. In the former case, the following equidistribution result is established.\n\t\n\t\\begin{thm}[{cf.~\\cite[Thm.~1]{Randol}}]\n\t\t\\label{thm:Randol}\n\t\tSuppose $C$ is the smooth boundary of a compact convex subset of $\\R^{d}$ with non-empty interior, and assume its Gaussian curvature is everywhere positive. Let $\\mu$ be the volume measure on $C$, renormalized to be a probability measure. Then the probability measures $m_t$ on $\\T^{d}$ defined by\n\t\t\\begin{equation*}\n\t\t\tm_t(A)=\\mu(\\{x\\in \\R^{n}:tx+\\Z^{d}\\in A \\})\\;, \\quad A\\subset \\T^{d} \\emph{ Borel }\n\t\t\\end{equation*} equidistribute towards the Haar measure on $\\T^{d}$ as $t\\to\\infty$.\n\t\\end{thm} \n\t\n\t\\begin{rmk}\n\t\t\\label{rmk:quantitative}\n\t\t\\begin{itemize}\n\t\t\t\\item [(a)] The result in Theorem~\\ref{thm:Randol} is actually quantitative: for every sufficiently regular function $f$ on $\\T^{d}$, it holds that\n\t\t\t\\begin{equation*}\n\t\t\t\t\\biggl|\\int_{\\T^{d}}f\\;\\text{d}m_t-\\int_{\\T^{d}}f\\;\\text{d}m_{\\T^{d}}\\biggr|\\ll_{f,d}t^{-(d-1)\/2}\\;,\n\t\t\t\\end{equation*}\n\t\t\twhere $m_{\\T^d}$ denotes the Haar probability measure on $\\T^{d}$.\n\t\t\t\\item [(b)] As a special case of Theorem~\\ref{thm:Randol}, expanding spheres in $\\R^{d}$ equidistribute on the standard torus with a polynomial rate depending on the dimension $d$. The study of expanding spheres in other geometric settings shall be a driving theme of this manuscript. \n\t\t\t\\item [(c)] In~\\cite[Thm.~2]{Randol}, equidistribution is generalized to uniform measures supported on lower-dimensional rectilinear simplices in $\\R^{d}$.\n\t\t\\end{itemize}\n\t\\end{rmk}\n\t\n\tThe proof of Theorem~\\ref{thm:Randol} (more generally, of its quantitative version stated in Remark~\\ref{rmk:quantitative}(a)) relies on classical Fourier analysis on the $d$-dimensional torus; it is remarkably straightforward, once the decay properties at infinity of the Fourier transform of the measure $\\mu$\n\tare known. As such, it has been elaborated upon by Strichartz in~\\cite{Strichartz} to prove the following generalization of Theorem~\\ref{thm:Randol}. Let us say that the Fourier transform $\\hat{\\mu}\\colon \\R^{d}\\to \\C$ of $\\mu$ decays on rays if \n\t\\begin{equation}\n\t\t\\label{eq:raydecay}\n\t\t\\lim\\limits_{t\\to\\infty}\\hat{\\mu}(tx)=0\n\t\\end{equation}\n\tfor every nonzero vector $x\\in \\R^{d}$.\n\t\\begin{thm}[{cf.~\\cite[Lem.~1]{Strichartz}}]\n\t\t\\label{thm:Strichartz}\n\t\tLet $\\mu$ be a Borel probability measure on $\\R^{d}$ whose Fourier transform decays on rays. Then the conclusion of Theorem~\\ref{thm:Randol} holds true. \n\t\\end{thm}\n\t\n\tIt actually suffices that $\\hat{\\mu}$ decays on integral rays, that is, that~\\eqref{eq:raydecay} is verified, less restrictively, for any non-zero $x\\in \\Z^{d}$. On the other hand, it turns out (cf.~\\cite[Lem.~1]{Strichartz}) that decay on arbitrary rays is equivalent to equidistribution of the projections of the $\\mu_t$ onto any torus $\\R^{d}\/\\La$, where $\\La$ is a lattice in $\\R^{d}$.\n\t\n\t\n\t\n\tTo conclude this brief account of the state of the art on the problem in flat geometry,  we mention that the question of the limiting distribution of expanding circles has been recently examined in the setting of translation surfaces by Colognese and Pollicott~\\cite{Colognese-Pollicott}, who prove (non-effective) equidistribution towards a probability measure which is equivalent, though in general not proportional, to the area measure on the given surface.\n\t\n\t\n\t\\subsubsection*{The hyperbolic case}\n\tAs already mentioned, Randol's investigations in~\\cite{Randol} were not confined to the zero-curvature case. In the hyperbolic framework, namely when the sectional curvature is constantly equal to $-1$, $S$ is a compact connected hyperbolic $d$-manifold ($d\\geq 2$) and $N$ can be taken as its Riemannian universal covering manifold, that is, the $d$-dimensional hyperbolic space $\\Hyp^{d}$. A choice of the homotheties $(h_t)_{t>0}$ is determined by fixing a base point $x_0\\in \\Hyp^{d}$ and letting $h_t$ be the map which transforms\\footnote{This produces a well-defined assignment, as $\\Hyp^{d}$ is a uniquely geodesic metric space (cf.~\\cite[Part I, Chap.~1]{Bridson-Haefliger}) for the distance induced by the hyperbolic Riemannian metric.} each Riemannian geodesic $\\gamma(s)$ ($s\\in \\R$) passing through $x_0$ at time $0$ into the geodesic $\\gamma(ts)$. In this context, the result that can be elicited from the discussion in~\\cite{Randol} reads as follows.\n\t\\begin{thm}[{cf.~\\cite{Randol}}]\n\t\t\\label{thm:Randolhyp}\n\t\tLet $S$ be a compact connected hyperbolic $d$-manifold, $\\pi\\colon \\Hyp^{d}\\to S$ the universal covering map, $C$ a $(d-1)$-dimensional hyperbolic  sphere of unit radius centered at a point $x_0\\in \\Hyp^{d}$, $\\mu$ the unique isometrically-invariant\\footnote{Here we obviously intend invariance under isometries of $C$ equipped with the induced hyperbolic metric.} Borel probability measure on $C$, $(h_t)_{t>0}$ the family of homotheties $\\Hyp^{d}\\to \\Hyp^{d}$ with center $x_0$ defined as above. Then the probability measures $m_t$ on $S$ defined by\n\t\t\\begin{equation}\n\t\t\t\\label{eq:measureshyperbolic}\n\t\t\tm_t(A)=\\mu(\\{x\\in \\Hyp^{d}:\\pi\\circ h_t(x)\\in A \\})\\;, \\quad A\\subset S \\emph{ Borel}\n\t\t\\end{equation}\n\t\tequidistribute towards the renormalized volume measure on $S$ as $t\\to\\infty$.\n\t\\end{thm}\n\t\n\t\\begin{rmk}\n\t\tHere again the result takes on a quantitative form: the rate of equidistribution of the measures $m_t$ defined in~\\eqref{eq:measureshyperbolic} is exponential, as opposed to the Euclidean case, with the exponent depending on the spectral gap of the hyperbolic manifold $S$ (cf.~\\cite{Randol}).\n\t\\end{rmk}\n\t\n\tAkin in spirit to the proof of Theorem~\\ref{thm:Randol}, the argument leading to Theorem~\\ref{thm:Randolhyp} is based upon the harmonic analysis of locally symmetric spaces via techniques related to the Selberg trace formula; for those, the reader is referred to Selberg's original article~\\cite{Selberg-trace}.\n\t\n\t\n\t\n\t\\subsubsection*{Lifting the question to unit tangent bundles} \n\t\n\tLet us now consider the following upgraded version of the problem explored in the foregoing paragraphs. Suppose that $S$, $N$ and $(h_t)_{t>0}$ are as in the beginning of the present section, with $y_0\\in N$ being the common center of the homotheties $h_t$, and let $C$ be a compact Riemannian hypersurface in $N$.  Assume further that the Riemannian distance function on $N$ turns it into a uniquely geodesic metric space\\footnote{This is certainly the case for $N=\\R^d$ and $N=\\Hyp^{d}$, equipped respectively with the standard Euclidean metric and with the hyperbolic metric.}. If $T^{1}N$ denotes the unit tangent bundle of $N$, then $C$ identifies uniquely the subset $\\tilde{C}$ of $T^{1}N$ consisting of all pairs $(x,v)$, where $x$ is a point in $C$ and $v$ is the unit-length tangent vector to the unique geodesic connecting $y_0$ to $x$. Similarly, if $C_t$ indicates the image of $C$ under the homothety $h_t$ for any $t>0$, we denote by $\\tilde{C_t}$ its lift to $T^{1}N$ obtained in the previous fashion. A natural question thus arises as to the asymptotic distribution of the $\\tilde{C_t}$ when projected to the unit tangent bundle of $S$; in this respect, the natural choice of a measure on $\\tilde{C_t}$ is given by the push-forward of the renormalized volume measure on $C_t$ under the canonical identification of the latter submanifold with its lift $\\tilde{C_t}$. If the projections to $S$ of the hypersurfaces $C_t$ equidistribute towards the normalized volume measure on $S$, it may be expected that the projections of the lifts $\\tilde{C_t}$ equidistribute towards the corresponding Liouville measure on the unit tangent bundle $T^{1}S$ (cf.~\\cite[Part 1, Sec.~5.4]{Hasselblatt-Katok}). \n\t\n\tThe question lends itself to a description in the language of smooth dynamical systems. If $(g_t^{(N)})_{t\\in \\R}$ is the geodesic flow on $T^{1}N$ (cf.~\\cite[Part 1, Chap.~5]{Hasselblatt-Katok}), it takes a few moments to realize that, for any $t>1$, the set $\\tilde{C_t}$ is the image of the original lift $\\tilde{C}$ under the transformation $g_{t-1}^{(N)}$; the same relation holds for the natural measures carried by the latter sets, and carries over to their projections to $T^1S$, using the geodesic flow $(g_t^{(S)})_{t\\in \\R}$ defined on it in place of $(g_t^{(N)})_{t\\in \\R}$. \n\t\n\tThe equidistribution problem in this formulation is treated in Margulis' thesis~\\cite{Margulis-thesis}, which contains several striking developments and applications of the theory of Anosov systems to the large-scale geometry of negatively curved manifolds; among those, a proof is provided of equidistribution, towards the Liouville measure, of lifts of expanding circles on finite-volume hyperbolic surfaces. \n\n\tFor further comments thereupon, as well as for the connection to the hyperbolic circle problem, the reader is referred to Section~\\ref{sec:circleproblemhyperbolic}.\n\t\n\tIt is the chief purpose of the present work to provide a precise asymptotic expansion for the equidistribution rate of lifts of dilating hyperbolic circles, as well as of arbitrary sub-arcs thereof, on unit tangent bundles of compact hyperbolic surfaces; in the vein of the works of Randol~\\cite{Randol} and Strichartz~\\cite{Strichartz}, we resort to a spectral approach originating in the work of Ratner~\\cite{Ratner} on quantitative mixing of geodesic and horocycle flows on Riemann surfaces of finite volume. Section~\\ref{sec:quantitativeequidistribution} describes such results and expands on their connection to previous developments, whereas Sections~\\ref{sec:introductionCLT} and~\\ref{sec:circleproblemhyperbolic} discuss a number of applications to statistical limit theorems and the hyperbolic lattice point counting problem. \n\t\n\tWe conclude this historical overview by pointing out that    \n\tMargulis' groundbreaking contributions in~\\cite{Margulis-thesis}, together with the gradual emergence of conspicuous applications to counting and Diophantine problems, spawned intensive research aimed at understanding the asymptotic distribution properties of translates of finite-volume subgroup orbits, as well as of more general subsets, on homogeneous spaces\\footnote{It is worth noticing at this point that lifts of expanding hyperbolic circles represent a particular instance, as they are geodesic translates of orbits of the maximal compact subgroup $\\SO_2(\\R)$ on quotients of $\\SL_2(\\R)$: see Section~\\ref{sec:quantitativeequidistribution}.}; without purporting to provide an exhaustive list, we mention in this direction the works~\\cite{Benoist-Oh,Einsiedler-Margulis-Venkatesh,Eskin-McMullen,Eskin-Mozes-Shah,Kra-Shah-Sun,Shah,Shah-second,Shah-third,Shah-fourth,PYang}. \n\t\n\t\\subsection{The setup: circles in hyperbolic surfaces and in their unit tangent bundles}\n\t\n\tWe now set the stage for the main questions we address in the present manuscript, referring to Section~\\ref{sec:preliminaries} for the required background.\n\tLet $\\Gamma<\\SL_2(\\R)$ be a cocompact lattice, that is, a discrete subgroup of $\\SL_2(\\R)$ such that the quotient space $\\Gamma\\bsl \\SL_2(\\R)$ is compact; we indicate the latter homogeneous space with $M$. The group $\\Gamma$ acts properly discontinuously and isometrically on the Poincar\\'{e} upper half-plane $\\Hyp=\\{z=x+iy \\in \\C:y>0 \\}$, endowed with the standard hyperbolic Riemannian metric, by M\\\"{o}bius transformations; when the projection of $\\Gamma$ to $\\PSL_2(\\R)=\\SL_2(\\R)\/\\{\\pm I_2\\}$ is torsion-free, the quotient $S=\\Gamma\\bsl \\Hyp$ is a compact connected orientable smooth surface, inheriting a hyperbolic metric from $\\Hyp$. With respect to such a metric, there is a canonical identification of $M$ with (possibly, a double cover of) the unit tangent bundle\\footnote{More precisely, if $\\Gamma$ is the preimage under the canonical projection map $\\SL_2(\\R)\\to \\PSL_2(\\R)$ of a cocompact lattice in $\\PSL_2(\\R)$, then $M$ identifies with $T^{1}S$; else, it is a double cover thereof. \n\t\t\n\t\tIn the case where the image of $\\Gamma$ in $\\PSL_2(\\R)$ has non-trivial torsion elements, then $S$ has the structure of an orbifold (cf.~\\cite[Chap.~13]{Ratcliffe}). For the purposes of the paper, we shall never be concerned with this distinction.} $T^{1}S$.\n\t\n\tLet $(r_{s})_{s\\in \\R}$ be the one-parameter flow on $M$ defined by \n\t\\begin{equation}\n\t\t\\label{eq:rotationflow}\n\t\tr_s(\\Gamma g)=\\Gamma g \\begin{pmatrix}\n\t\t\t\\cos{s\/2}&\\sin{s\/2}\\\\\n\t\t\t-\\sin{s\/2}& \\cos{s\/2}\n\t\t\\end{pmatrix}\n\t\t=\\Gamma g \\exp{s\\Theta}, \\quad \\Theta=\\begin{pmatrix}\n\t\t\t0&1\/2\\\\\n\t\t\t-1\/2&0\n\t\t\\end{pmatrix}\n\t\t\\in \\sl_2(\\R)=\\operatorname{Lie}(\\SL_2(\\R)),\n\t\\end{equation}\n\tand denote by $(\\phi_t^{X})_{t\\in \\R}$ the geodesic flow on $M$, which is given algebraically by \n\t\\begin{equation}\n\t\t\\label{eq:geodesicflow}\n\t\t\\phi^{X}_{s}(\\Gamma g)=\\Gamma g \\begin{pmatrix} e^{t\/2}&0\\\\ 0&e^{-t\/2} \\end{pmatrix}=\\Gamma g \\exp{tX},\\quad X=\\begin{pmatrix}\n\t\t\t1\/2 &0\\\\\n\t\t\t0&-1\/2 \\end{pmatrix}\n\t\t\\in \\sl_2(\\R).\n\t\\end{equation}\n\t\n\tFor any point $p=\\Gamma g\\in M$, the orbit of $p$ under the flow $(r_s)_{s\\in \\R}$ is the preimage of $z=\\pi(p)$ under the fibration $\\pi\\colon M\\to S$. Therefore, if $M$ identifies with $T^{1}S$, then this set consists of all unit tangent vectors to $z\\in S$. For any real number  $t>0$, the time-$t$ geodesic evolution  $\\phi_t^{X}(\\{r_s(p):s\\in \\R \\})$ of the previous set coincides with the projection to $M$ of the subset of $T^{1}\\Hyp$ given by all outward-pointing normal vectors to the hyperbolic circle in $\\Hyp$ of radius $t$ centered at (a fixed representative in $\\Hyp$ of) $z$.\n\t\n\tWe indicate with $\\vol$ the Haar probability measure on $M$, that is, the unique $\\SL_2(\\R)$-invariant Borel probability measure on $M$; under the identification of $M$ with $T^1S$, it coincides with the Liouville measure projecting to the normalized hyperbolic area measure on $S$. For any $r\\in \\N\\cup \\{\\infty \\}$, we denote by $\\mathscr{C}^{r}(M)$ the set of complex-valued functions of class $\\cC^{r}$ defined on the smooth manifold $M$. The supremum norm of a continuous function $f\\colon M\\to \\C$ is denoted by $\\norm{f}_{\\infty}$. For any $m\\in \\N_{\\geq 1}$, $f\\in \\cC^{m}(M)$ and $j\\in \\{0,\\dots,m\\}$ let $\\nabla^{j}f$ be the $j^{\\text{th}}$ covariant derivative of $f$ and $|(\\nabla^{j}f)(p)|$ its norm at a point $p \\in M$. Define then the $\\cC^{m}$-norm of $f$ (cf.~\\cite[Chap.~1]{Aubin}) as\n\t\\begin{equation}\n\t\t\\label{eq:Cknorm}\n\t\t\\norm{f}_{\\cC^{m}}=\\sum_{j=0}^{m}\\sup\\limits_{p \\in M}|(\\nabla^{j}f)(p)|\\;.\n\t\\end{equation}\n\t\n\tLet $L^{2}(M)$ be the Hilbert space of complex-valued functions on $M$ whose modulus is square-integrable with respect to the measure $\\vol$, and denote by\n\t\\begin{equation*}\n\t\t\\langle \\phi,\\psi\\rangle=\\int_M \\phi\\;\\bar{\\psi}\\;\\text{d}\\vol\n\t\\end{equation*}\n\tthe inner product of two elements $\\phi,\\psi\\in L^{2}(M)$.\n\t\n\tDefine two additional elements\n\t\\begin{equation*}\n\t\tU=\n\t\t\\begin{pmatrix}\n\t\t\t0&1\\\\\n\t\t\t0&0\n\t\t\\end{pmatrix}\n\t\t\\;,\\quad \n\t\tV=\n\t\t\\begin{pmatrix}\n\t\t\t0&0\\\\\n\t\t\t1&0\n\t\t\\end{pmatrix}\n\t\\end{equation*}\n\tin the Lie algebra $\\sl_2(\\R)$.\n\tIdentifying elements of the universal enveloping algebra of $\\sl_2(\\R)$ with left-invariant differential operators on the space $\\cC^{\\infty}(M)$, we define the Casimir operator as the second-order differential operator $\\square=-X^{2}+X-UV\\colon \\cC^{2}(M)\\to \\cC^0(M)$. It admits a unique maximal extension to an unbounded self-adjoint operator on $L^{2}(M)$; in particular, its spectrum $\\text{Spec}(\\square)$ consists of real numbers. As $M$ is compact, it is well-known that $\\text{Spec}(\\square)$ is pure point, and is a discrete subset of $\\R$. The elements of $\\Spec(\\square)$ classify the irreducible representations strongly contained in the Koopman representation arising from the measure-preserving action of $\\SL_2(\\R)$ on the measure space $(M,\\vol)$, as belonging to the principal, complementary or discrete series representations if the corresponding eigenvalue $\\mu$ satisfies, respectively, $\\mu\\geq 1\/4$, $0<\\mu<1\/4$, $\\mu\\leq 0$. With the normalization we have chosen\\footnote{Since $\\sl_2(\\R)$ is a simple Lie algebra, Casimir elements in its universal enveloping algebra are uniquely determined up to real scalars.}, the action of $\\square$ on $\\cC^{2}$-functions defined on the surface $S$ is given by the Laplace-Beltrami operator $\\Delta_S$ associated to the hyperbolic structure on $S$.\n\t\n\tWe are interested in quantitative equidistribution properties of the uniform probability measures supported on the circle arcs\n\t\\begin{equation*}\n\t \\phi_t^{X}(\\{r_s(p):0\\leq s\\leq \\theta \\})\n\t \\end{equation*}\n \tas $t$ goes to infinity, for every fixed $p \\in M$ and $\\theta\\in (0,4\\pi]$ (cf.~our normalization of $\\Theta$ in~\\eqref{eq:rotationflow}, a full circle corresponds to $\\theta=4\\pi$). For any parameter $\\theta\\in (0,4\\pi]$ and any continuous function $f\\colon M\\to \\C$, we thus define the function $k_{f,\\theta}\\colon M\\times \\R\\to \\C$ as\n\t\\begin{equation}\n\t\t\\label{eq:ktheta}\n\t\tk_{f,\\theta}(p,t)\\coloneqq \\frac{1}{\\theta} \\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\\; ,\\quad p \\in M,\\;t\\in \\R.\n\t\\end{equation}\n\tIn the forthcoming subsection, we provide a precise asymptotic expansion of $k_{f,\\theta}(p,t)$ as $t$ tends to infinity, first for joint eigenfunctions\\footnote{As we shall explain in Section~\\ref{sec:unitaryrepresentations}, there exists an orthonormal basis of $L^{2}(M)$ consisting of such joint eigenfunctions; Theorem~\\ref{thm:case_Theta_eigenfn} is thus to be regarded as a building block for the more general Theorem~\\ref{thm:mainexpandingtranslates}.} of the operators $\\square$ and $\\Theta$ (Theorem~\\ref{thm:case_Theta_eigenfn}), and then for arbitrary functions fulfilling a suitable regularity condition (Theorem~\\ref{thm:mainexpandingtranslates}).\n\t\n\t\n\t\n\t\\subsection{Quantitative equidistribution of expanding translates of circle arcs}\n\t\\label{sec:quantitativeequidistribution}\n\t\n\tWe begin with the case of joint eigenfunctions of the operators $\\square$ and $\\Theta$.\n\tObserve that the left-invariant vector field $\\Theta\\in \\sl_2(\\R)$ acts as an unbounded skew-symmetric operator on $L^{2}(M)$; if $\\Theta f=\\lambda f$ for some $f\\in \\mathscr{C}^{1}(M)$ and $\\lambda \\in \\C$, then $\\lambda=\\frac{i}{2}n$ for some $n\\in \\Z$. \n\t\n\tIn the following statement and throughout, we associate to each $\\mu\\in \\text{Spec}(\\square)$\n\t the unique complex number $\\nu\\in \\R_{\\geq 0}\\cup i\\R_{>0}$ satisfying $1-\\nu^2=4\\mu$.\n\t\n\t\n\t\\begin{thm}\\label{thm:case_Theta_eigenfn}\n\t\tThere exist real constants $\\kappa_0$ and $\\kappa(\\mu)$ for any positive Casimir eigenvalue $\\mu$ such that the following assertions hold. Let $\\theta\\in (0,4\\pi]$, $\\mu\\in \\emph{Spec}(\\square)$ and $n\\in \\Z$, and suppose $f\\in \\mathscr{C}^{2}(M)$ satisfies $\\square f=\\mu f$, $\\Theta f=\\frac{i}{2}nf$. Define $k_{f,\\theta}(p,t)$ as in~\\eqref{eq:ktheta}.  \n\t\t\\begin{enumerate}\n\t\t\t\\item If $\\mu>1\/4$, there exist H\\\"{o}lder-continuous functions $D^{+}_{\\theta,\\mu,n}f,D^{-}_{\\theta,\\mu,n}f\\colon M\\to \\C$ with H\\\"{o}lder exponent $1\/2$ and\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f}_{\\infty}\\leq \\frac{\\kappa(\\mu)}{\\theta}(n^2+1)\\norm{f}_{\\mathscr{C}^{1}} \n\t\t\t\\end{equation*}\n\t\t\tsuch that, for every $p \\in M$ and $t\\geq 1$, \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estabovequarter}\n\t\t\t\tk_{f,\\theta}(p,t)=e^{-\\frac{t}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}D^{+}_{\\theta,\\mu,n}f(p)+e^{-\\frac{t}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}D^{-}_{\\theta,\\mu,n}f(p)+\\mathcal{R}_{\\theta,\\mu,n}f(p,t)\\;,\n\t\t\t\\end{equation}\n\t\t\twhere \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estremainderabovequarter}\n\t\t\t\t|\\mathcal{R}_{\\theta,\\mu,n}f(p,t)|\\leq \\frac{8\\kappa_0(n^{2}+1)}{\\theta\\; \\Im{\\nu}}\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;.\n\t\t\t\\end{equation}\n\t\t\t\\item If $\\mu=1\/4$, there exist H\\\"{o}lder-continuous functions $D^{+}_{\\theta,1\/4,n}f\\colon M\\to \\C$, with H\\\"{o}lder exponent $1\/2-\\varepsilon$ for every $\\varepsilon>0$, and $D^{-}_{\\theta,1\/4,n}f\\colon M\\to \\C$, with H\\\"{o}lder exponent $1\/2$, and satisfying\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{D^{\\pm}_{\\theta,1\/4,n}f}_{\\infty}\\leq \\frac{\\kappa(1\/4)}{\\theta}(n^2+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\t\t\t\\end{equation*}\n\t\t\tsuch that, for every $p \\in M$ and $t\\geq 1$,\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estquarter}\n\t\t\t\tk_{f,\\theta}(p,t)=e^{-\\frac{t}{2}}D^{+}_{\\theta,1\/4,n}f(p)+te^{-\\frac{t}{2}}D^{-}_{\\theta,1\/4,n}f(p)+\\mathcal{R}_{\\theta,1\/4,n}f(p,t)\\;,\n\t\t\t\\end{equation}\n\t\t\twhere \n\t\t\t\\begin{equation*}\n\t\t\t\t|\\mathcal{R}_{\\theta,1\/4,n}f(p,t)|\\leq \\frac{4\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}(t+1)e^{-t}\\;.\n\t\t\t\\end{equation*}\n\t\t\t\\item If $0<\\mu<1\/4$, there exist functions $D^{+}_{\\theta,\\mu,n}f, D^{-}_{\\theta,\\mu,n}f \\colon M\\to \\C$, respectively H\\\"{older}-continuous with H\\\"{o}lder exponent $\\frac{1-\\nu}{2}$ and  of class $\\cC^{1}$, and satisfying\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f}_{\\infty}\\leq \\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\t\t\t\\end{equation*}\n\t\t\tsuch that, for every $p \\in M$ and $t\\geq 1$,\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estpositivebelowquarter}\n\t\t\t\tk_{f,\\theta}(p,t)=e^{-\\frac{1+\\nu}{2}t}D^{+}_{\\theta,\\mu,n}f(p)+e^{-\\frac{1-\\nu}{2}t}D^{-}_{\\theta,\\mu,n}f(p)+\\mathcal{R}_{\\theta,\\mu,n}f(p,t)\\;,\n\t\t\t\\end{equation}\n\t\t\twhere \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estremainderbelowquarter}\n\t\t\t\t|\\mathcal{R}_{\\theta,\\mu,n}f(p,t)|\\leq \\frac{4\\kappa_0}{\\theta\\nu(1-\\nu)(1+\\nu)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;.\n\t\t\t\\end{equation}\n\t\t\t\\item If $\\mu=0$, there exists a function $G_{\\theta,n}f\\colon M\\times \\R_{>0}\\to \\C$, with $G_{\\theta,n}f(\\cdot,t)$ of class $\\cC^{1}$ for any $t>0$, $G_{\\theta,n}f(p,\\cdot)$ continuous for every $p \\in M$ and\n\t\t\t\\begin{equation*}\n\t\t\t\t\\sup\\limits_{p\\in M,\\;t>0}|G_{\\theta,n}f(p,t)|\\leq \\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\cC^{1}}\n\t\t\t\\end{equation*}\n\t\t\tsuch that, for every $p \\in M$ and $t\\geq 1$,  \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estzero}\n\t\t\t\tk_{f,\\theta}(p,t)=\\int_{M}f\\;\\emph{d}\\vol+e^{-t}\\int_1^{t}-G_{\\theta,n}f(p,\\xi)\\;\\emph{d}\\xi+\\mathcal{R}_{\\theta,0,n}f(p,t)\\;,\n\t\t\t\\end{equation}\n\t\t\twhere \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:remainderzero}\n\t\t\t\t|\\mathcal{R}_{\\theta,0,n}f(p,t)|\\leq\\frac{8e\\pi+\\kappa_0}{\\theta} (n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;.\n\t\t\t\\end{equation}\n\t\t\t\\item If $\\mu<0$ then, for every $p \\in M$ and $t\\geq 1$,\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:estimatediscreteseries}\n\t\t\t\t|k_{f,\\theta}(p,t)|\\leq \\frac{1}{\\theta}\\biggl(\\frac{4\\kappa_0}{(\\nu-1)(\\nu+1)}+\\frac{2e\\pi(3+\\nu)}{\\nu}\\biggr)(n^{2}+1) \\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\t\\end{thm}\n\t\n\t\n\t\\begin{rmk}\n\t\t\tThe asymptotic expansions for $k_{f,\\theta}(p,t)$ are easily transferred to the case of large negative values of the time parameter $t$, by noticing that\n\t\t\t\\begin{equation*}\n\t\t\t\tk_{f,\\theta}(p,-t)=k_{f \\circ r_{\\pi},\\theta}(r_{\\pi}(p),t) \n\t\t\t\\end{equation*}\n\t\tfor any $p \\in M$ and $t\\geq 1$.\n\t\\end{rmk}\n\t\n\tTaking advantage of Theorem~\\ref{thm:case_Theta_eigenfn} and of standard harmonic analysis on the group $\\SL_2(\\R)$, we establish an asymptotic expansion of $k_{f,\\theta}(p,t)$ for all sufficiently regular, but otherwise arbitrary test functions $f$.\n\tDefine the Laplacian on $M$ to be the second-order linear differential operator $\\Delta=\\square-2\\Theta^{2}$. \n\tFor any $s\\in \\R_{>0}$, let $W^{s}(M)$ be the Sobolev space of order $s$ on the manifold $M$, that is, the Hilbert-space completion of the complex vector space $\\cC^{\\infty}(M)$ of smooth functions on $M$ endowed with the inner product \n\t\\begin{equation*}\n\t\t\\langle \\phi,\\psi \\rangle_{W^s}=\\langle (1+\\Delta)^{s}\\phi,\\psi\\rangle\\;,\\quad \\phi,\\psi\\in \\cC^{\\infty}(M).\n\t\\end{equation*}\n\t\n\tAs $M$ is compact, the well-known Sobolev Embedding Theorem (which we recall in Theorem~\\ref{thm:Sobolevembedding}) ensures the existence of a continuous embedding $W^{s}(M)\\hookrightarrow \\cC^{r}(M)$ whenever $s\\in \\R_{>0}$ and $r\\in \\N$ are such that $s>r+3\/2$; explicitly, there is a constant $C_{r,s}\\in \\R_{>0}$ (which for definiteness we take equal to the operator norm of the corresponding embedding) such that $\\norm{f}_{\\cC^{r}}\\leq C_{r,s}\\norm{f}_{W^{s}}$ for any $f\\in W^{s}(M)$. Hereinafter, an element $f\\in W^{s}(M)$ for $s>3\/2$ is always identified with its unique continuous representative.\n\t\n\tSet\n\t\\begin{equation}\n\t\t\\label{eq:epszero}\n\t\t\\varepsilon_{0}=\n\t\t\\begin{cases}\n\t\t\t1 &\\text{ if }1\/4\\in \\text{Spec}(\\square)\\;,\\\\\n\t\t\t0 &\\text{ if }1\/4\\notin \\text{Spec}(\\square)\\;.\\\\\n\t\t\\end{cases}\n\t\\end{equation}\n\t\n\t\\begin{thm}\n\t\t\\label{thm:mainexpandingtranslates}\n\t\tThere exist real constants $C_{\\emph{Spec}}$ and $\\;C_{\\emph{Spec}}'$, depending only on the spectrum of the Casimir operator on $L^{2}(M)$, such that the following holds. Let $\\theta\\in (0,4\\pi]$ and $s>11\/2$ be real numbers, and suppose $f\\in W^{s}(M)$.\n\t\tThen there exist continuous functions $D^{+}_{\\theta,\\mu}f,\\;D^{-}_{\\theta,\\mu}f\\colon M\\to \\C$ for any positive Casimir eigenvalue $\\mu$, with\n\t\t\\begin{equation}\n\t\t\t\\label{eq:boundDthetamu}\n\t\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)\\cap \\R_{>0}}\\norm{D^{+}_{\\theta,\\mu}f}_{\\infty}+\\norm{D^{-}_{\\theta,\\mu}f}_{\\infty}\\leq \\frac{C_{\\emph{Spec}}'C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}},\n\t\t\\end{equation}\n\t\tsuch that, for every $p\\in M$ and $t\\geq 1$,\n\t\t\\begin{equation}\n\t\t\t\\label{eq:asymptoticgeneral}\n\t\t\t\\begin{split}\n\t\t\t\t\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_{t}\\circ r_s(p)\\;\\emph{d}s=&\\int_{M}f\\;\\emph{d}\\vol\\\\\n\t\t\t\t&+e^{-\\frac{t}{2}}\\biggl( \\sum_{\\mu\\in \\emph{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)} D^{+}_{\\theta,\\mu}f(p)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)} D^{-}_{\\theta,\\mu}f(p)\\biggr)\\\\\n\t\t\t\t&+\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}t}D^{+}_{\\theta,\\mu}f(p)+e^{-\\frac{1-\\nu}{2}t}D^{-}_{\\theta,\\mu}f(p)\\\\\n\t\t\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{t}{2}}D^{+}_{\\theta,1\/4}f(p)+te^{-\\frac{t}{2}}D^{-}_{\\theta,1\/4}f(p)\\bigr)+\\mathcal{R}_{\\theta}f(p,t)\\;,\n\t\t\t\\end{split}\n\t\t\\end{equation}\n\t\twhere\n\t\t\\begin{equation}\n\t\t\t\\label{eq:globalremainderestimate}\n\t\t\t|\\mathcal{R}_{\\theta}f(p,t)|\\leq \\frac{C_{\\emph{Spec}}C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}}(t+1)e^{-t} \\;.\n\t\t\\end{equation}\n\t\\end{thm}   \n\t\n\t\n\tWe record here below the ensuing effective equidistribution statement, in which we single out the two main terms of the asymptotic expansion, thereby highlighting the dependence of the latter on the spectral gap of the underlying hyperbolic surface $S=\\Gamma\\bsl \\Hyp$, defined as \n\t\\begin{equation*}\n\t\t\\mu_*=\\inf(\\text{Spec}(\\square)\\cap \\R_{>0})=\\inf(\\text{Spec}(\\Delta_S)\\setminus\\{0\\})\\;;\n\t\\end{equation*}\n\tits corresponding parameter is denoted by $\\nu_*$. \n\t\n\t\\begin{term}\n\t\tWe adopt the classical Landau notation $o(\\eta(t))$ for $\\eta\\colon \\R_{> 0}\\to \\R_{>0}$ tending to zero at infinity, to indicate a function $\\lambda\\colon \\R_{>0}\\to \\C$ such that $|\\lambda(t)|\/\\eta(t)\\to 0$ as $t\\to\\infty$. \n\t\\end{term}\n\t\n\t\n\t\\begin{cor}\n\t\t\\label{cor:effective}\n\t\tLet $\\theta,s,C_{1,s-3},C'_{\\emph{Spec}}$ and $f$ be as in Theorem~\\ref{thm:mainexpandingtranslates}. Then there exists a function $D^{\\emph{main}}_{\\theta}f\\colon M\\to \\C$ with\n\t\t\\begin{equation}\n\t\t\t\\label{eq:boundDthetamain}\n\t\t\t\\norm{D^{\\emph{main}}_{\\theta}f}_{\\infty}\\leq \\frac{C'_{\\emph{Spec}}C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}}\n\t\t\\end{equation}\n\t\tsuch that, for every $p \\in M$ and $t\\geq 1$, \n\t\t\\begin{equation}\n\t\t\t\\label{eq:effective}\n\t\t\t\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\emph{d}s=\\int_{M}f\\;\\emph{d}\\vol+\\;D^{\\emph{main}}_{\\theta}f(p)\\;t^{\\varepsilon_0}e^{-\\frac{1-\\Re{\\nu_*}}{2}t}+o(e^{-\\frac{1-\\Re{\\nu_*}}{2}t})\\;.\n\t\t\\end{equation}\n\t\\end{cor}\n\t\n\t\\begin{rmk}\n\t\tUsing the result of Dyatlov, Faure and Guillarmou in~\\cite{Dyatlov-Faure-Guillarmou} we can relate the coefficients $D^{-}_{\\theta,\\mu}f$, for $0< \\mu <1\/4$, to the resonant and co-resonant states in the first band for the geodesic flow (in other words, with the invariant distributions for the unstable and stable horocycle flow, respectively). \n\t\tLet $u_\\mu$ be a resonant state for the geodesic flow associated to the Pollicott-Ruelle resonance $-\\frac{1-\\nu}{2} \\in (-1\/2,0)$, and let $u^\\ast_\\mu$ be its dual (we refer to \\cite[Sec.~$1$ and $2$]{Dyatlov-Faure-Guillarmou} for the relevant definitions). \n\t\tIn the language of Flaminio-Forni's work~\\cite{Flaminio-Forni}, $u_\\mu$ is an invariant distribution for the unstable horocycle flow, and $u^\\ast_\\mu$ is an invariant distribution for the stable horocycle flow.\n\t\tBy virtue of \\cite[Thm.~4]{Dyatlov-Faure-Guillarmou}, we have, for any sufficiently small $\\varepsilon>0$ and for arbitrary $f,g \\in \\mathscr{C}^\\infty(M)$,\n\t\t\\begin{equation*}\n\t\t\t\\langle f \\circ \\phi^X_t, g\\rangle = \\int_{M}f\\;\\text{d}\\vol  \\int_{M}\\overline{g}\\;\\text{d}\\vol + \\sum_{\\mu \\in \\text{Spec}(\\square)\\cap (0,1\/4)} \\langle f, u_\\mu \\rangle \\, \\langle u_\\mu^\\ast , g \\rangle \\, e^{-\\frac{1-\\nu}{2}t} + O\\left( e^{-\\bigl(\\frac{1}{2}-\\varepsilon\\bigr)t}\\right).\n\t\t\\end{equation*}\n\t\tThus,\n\t\t\\begin{equation*}\n\t\t\t\\begin{split}\n\t\t\t\t&\\langle k_{f, \\theta}(\\cdot, t), g\\rangle = \\frac{1}{\\theta} \\int_0^\\theta \\langle f \\circ \\phi^X_t, g \\circ r_{-s}\\rangle \\;\\text{d}s \\\\\n\t\t\t\t& \\quad = \\int_{M}f\\;\\text{d}\\vol  \\int_{M}\\overline{g}\\;\\text{d}\\vol + \\sum_{\\mu \\in \\text{Spec}(\\square)\\cap (0,1\/4)} \\left(\\frac{1}{\\theta} \\int_0^\\theta \\langle f, u_\\mu \\rangle \\, \\langle u_\\mu^\\ast , g \\circ r_{-s}\\rangle \\;\\text{d}s \\right) \\, e^{-\\frac{1-\\nu}{2}t} + O\\left( e^{-\\bigl(\\frac{1}{2}-\\varepsilon\\bigr)t}\\right).\n\t\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tOn the other hand, by Theorem \\ref{thm:mainexpandingtranslates},\n\t\t\\begin{equation*}\n\t\t\t\\langle k_{f, \\theta}(\\cdot, t), g\\rangle = \\int_{M}f\\;\\text{d}\\vol  \\int_{M}\\overline{g}\\;\\text{d}\\vol + \\sum_{\\mu \\in \\text{Spec}(\\square)\\cap (0,1\/4)} \\langle D^{-}_{\\theta,\\mu}f, g \\rangle \\,  e^{-\\frac{1-\\nu}{2}t} + O\\left( e^{-\\bigl(\\frac{1}{2}-\\varepsilon\\bigr)t}\\right).\n\t\t\\end{equation*}\n\t\tEquating coefficients, we conclude that the functions $D^{-}_{\\theta,\\mu}f,\\; 0< \\mu <1\/4$, coincide as distributions with the corresponding averaged resonant states, namely\n\t\t\\begin{equation}\n\t\t\t\\label{eq:D-_resonant_states}\n\t\t\tD^{-}_{\\theta,\\mu}f = \\langle f, u_\\mu \\rangle \\cdot \\frac{1}{\\theta} \\int_0^\\theta u_\\mu^\\ast \\circ r_{s} \\;\\text{d}s\\;.\n\t\t\\end{equation}\n\t\tOf particular interest is the full average $\\theta = 4 \\pi$: in this case, $D^{-}_{4\\pi,\\mu}f$ is a multiple of $\\frac{1}{4\\pi} \\int_0^{4\\pi} u_\\mu^\\ast \\circ r_{s} \\;\\text{d}s$, which is an eigenfunction of the Laplacian of eigenvalue $\\mu$, see \\cite[p.~931]{Dyatlov-Faure-Guillarmou}.\n\t\t\n\t\tFrom \\eqref{eq:D-_resonant_states}, we also deduce that $D^{-}_{\\theta,\\mu}f $ is identically zero if and only if $\\langle f, u_\\mu \\rangle =0$, or, in other words, if and only if $f$ lies in the kernel of the invariant distribution for the unstable horocycle flow with Casimir parameter $\\mu$ (see \\cite[Sec.~3]{Flaminio-Forni}).\n\t\t\n\t\tIn order to prove an analogous relation between the other coefficients $D^{\\pm}_{\\theta,\\mu}f $ and the horocycle invariant distributions, we would need an asymptotic expansion of the correlations for the geodesic flow as in \\cite[Thm.~4.3]{Forni}, but with an explicit dependence of the coefficients in terms of the stable and unstable horocycle-invariant distributions.   \n\t\\end{rmk}\n\t\n\t\n\t\\begin{rmk}\n\t\t\\label{rmk:equivalentmetrics}\n\t\tWe collect here below further comments about Theorem~\\ref{thm:case_Theta_eigenfn}, Theorem~\\ref{thm:mainexpandingtranslates} and Corollary~\\ref{cor:effective}.\n\t\t\\begin{enumerate}\n\t\t\\item The H\\\"{o}lder-continuity claims concerning the coefficients $D^{\\pm}_{\\theta,\\mu,n}$ appearing in Theorem~\\ref{thm:case_Theta_eigenfn} tacitly involve the choice of a distance function $d$ on $M$. It is intended that $d$ is the Riemannian distance function induced on the connected manifold $M$ by a Riemannian metric $g$. The H\\\"{o}lder-continuity property, as well as the H\\\"{o}lder exponent, of $D^{\\pm}_{\\theta,\\mu,n}$ is independent of the choice of such a $g$, as any two Riemannian metrics on a compact manifold induce Lipschitz-equivalent metrics (see \\cite[Lem.~13.28]{Lee}).\n\t\t\\item We point out the analogy of Theorem~\\ref{thm:case_Theta_eigenfn} with the asymptotics of horocycle ergodic integrals for Casimir eigenfunctions established in~\\cite[Thm.~1]{Rav} (in the same way, Theorem~\\ref{thm:mainexpandingtranslates} mirrors~\\cite[Thm.~2]{Rav}). This similarity stems computationally from the application of the exact same spectral method in both circumstances; as a matter of fact, it comes as no surprise from a geometric standpoint, for large hyperbolic circles approximate orbits of the unstable horocycle flow. \n\t\t\\item For $\\theta=4\\pi$, we recover the qualitative equidistribution of expanding circles towards the uniform measure $\\vol$ obtained by Margulis in~\\cite{Margulis-thesis}. \n\t\t\\item The equidistribution rate in Corollary~\\ref{cor:effective} matches exactly the mixing rate of the geodesic flow on $M$ obtained by Ratner in~\\cite[Thm.~2]{Ratner}.\n\t\t\\item As suggested by the underlying geometric picture, the various upper bounds in the statements of Theorem~\\ref{thm:case_Theta_eigenfn} and~\\ref{thm:mainexpandingtranslates} indicate that the speed of equidistribution improves as $\\theta$ increases to $4\\pi$, that is, as the length of the initial circle arc gets larger. \n\t\t\\end{enumerate}\n\t\\end{rmk}\n\t\n\tAs a special case of Theorem \\ref{thm:mainexpandingtranslates}, we obtain an asymptotic expansion for the equidistribution rate of $\\Theta$-invariant functions; in other words, we provide the precise asymptotic behaviour of large circles on the compact hyperbolic surface $S$, identified, here and afterwards whenever convenient, with the double coset space $\\Gamma\\bsl \\SL_2(\\R)\/\\SO_2(\\R)$.\n\t\n\tLet $d_{\\Hyp}$ denote the distance function on $\\Hyp$ arising from the hyperbolic Riemannian metric. For any $z\\in \\Hyp$ and $t>0$, let $C_{\\Hyp}(z,t)=\\{z'\\in \\Hyp:d_{\\Hyp}(z,z')=t \\}$ be the hyperbolic circle of radius $t$ centered at $z$, and denote by $C_{S}(z,t)$ its projection to $S$. With $m_{C_S(z,t)}$ we indicate the projection to $S$ of the unique isometrically-invariant Borel probability measure supported on $C_{\\Hyp}(z,t)$. Finally, let $m_{S}$ be the hyperbolic area measure on $S$, normalized to be a probability measure.\n\t\n\t\\begin{thm}\n\t\t\\label{thm:expandingonsurface}\n\t\tLet $\\theta\\in (0,4\\pi],\\;s>9\/2$, $\\tilde{f}\\colon S \\to \\C$ a function such that the $\\SO_2(\\R)$-invariant function $f\\colon M\\to \\C$ defined by $f(\\Gamma g)=\\tilde{f}(\\Gamma g \\SO_2(\\R))$ for any $g\\in \\SL_2(\\R)$ is in $W^{s}(M)$. Then, for  every $z=\\Gamma g\\SO_2(\\R)\\in S$,  $p=\\Gamma g \\in M$ and $t\\geq 1$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{split}\n\t\t\t\t\\int_{S}\\tilde{f}\\;\\emph{d}m_{C_S(z,t)}&=\\int_{S}\\tilde{f}\\;\\emph{d}m_S\n\t\t\t\t+e^{-\\frac{t}{2}}\\biggl( \\sum_{\\mu\\in \\emph{Spec}(\\Delta_S),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)} D^{+}_{4\\pi,\\mu}f(p)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)} D^{-}_{4\\pi,\\mu}f(p)\\biggr)\\\\\n\t\t\t\t&+\\sum_{\\mu\\in \\emph{Spec}(\\Delta_S),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}t}D^{+}_{4\\pi,\\mu}f(p)+e^{-\\frac{1-\\nu}{2}t}D^{-}_{4\\pi,\\mu}f(p)\\\\\n\t\t\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{t}{2}}D^{+}_{4\\pi,1\/4}f(p)+te^{-\\frac{t}{2}}D^{-}_{4\\pi,1\/4}f(p)\\bigr)+\\mathcal{R}_{4\\pi}f(p,t)\\;.\n\t\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tIn particular, the functions $\\mathcal{R}_{4\\pi}f$ and $D^{\\pm}_{4\\pi,\\mu}f,\\;\\mu\\in \\emph{Spec}(\\Delta_S)\\cap \\R_{>0}$, defined as in Theorem~\\ref{thm:mainexpandingtranslates}, are $\\SO_2(\\R)$-invariant. \n\t\\end{thm}  \n\t\n\tClearly, an analogous statement holds for arbitrary sub-arcs of the circles $C_{S}(z,t)$.\n\t\n\t\\begin{rmk}\n\t\tIt is worth highlighting that, in the case of an $\\SO_2(\\R)$-invariant observable $f$, we require mildly less restrictive assumptions on its Sobolev regularity; this will become relevant in Section~\\ref{sec:latticepoint} when we deal with the error rate for the hyperbolic lattice point counting problem. \n\t\t\n\t\tFurthermore, we emphasize that Theorem~\\ref{thm:expandingonsurface} improves upon the equidistribution results in~\\cite{Randol} (which, on the other hand, apply to any dimension) in a twofold way: it demands less restrictive conditions on the regularity of test functions and, more importantly, it refines Randol's upper bound on the equidistribution rate by giving a precise asymptotic expansion.\n\t\\end{rmk}\n\t\n\tLeveraging the explicit dependence of the asymptotics in Theorem~\\ref{thm:mainexpandingtranslates} on the upper bound $\\theta$ of the domain of parametrization of the circle arc under consideration, we deduce a sufficient quantitative condition for the equidistribution of shrinking pieces of expanding circles; this is in the vein of Str\\\"{o}mbergsson's results in~\\cite{Strombergsson-closedhorocycles}, where the analogous question is investigated for shrinking portions of closed horocycles on non-closed hyperbolic surfaces of finite volume.\n\t\n\t\\begin{cor}\n\t\t\\label{cor:shrinkingarcs}\n\t\tLet $p \\in M$, $\\theta_1,\\theta_2\\colon \\R_{>0} \\to (0,4\\pi)$ two functions with $\\theta_1(t)\\leq \\theta_2(t)$ for any $t>0$, and consider the circle sub-arcs $\\gamma_{t}=\\{\\phi^{X}_t\\circ r_{s}(p):\\theta_1(t)\\leq s \\leq \\theta_2(t)\\}$. For any $t>0$, let $\\mu_t$ be the normalized restriction to $\\gamma_t$ of the unique isometrically-invariant measure on the corresponding full circle. Assume that there exist $t_0>0$ and a function $\\eta\\colon \\R_{>0}\\to \\R_{>0}$ with $\\eta(t)\\to\\infty$ as $t\\to\\infty$ such that $\\theta_2(t)-\\theta_1(t)\\geq\\eta(t) e^{-\\frac{1-\\Re{\\nu_*}}{2}t}$ for any $t\\geq t_0$. Then the circle arcs $\\gamma_t$ equidistribute as $t\\to\\infty$: more precisely, the measures $\\mu_t$ converge in the weak$^*$ topology, as $t\\to\\infty$, towards the uniform measure $\\vol$ on $M$. \n\t\\end{cor} \n\t\n\t\\subsection{Statistical limit theorems for deviations from the average}\n\t\\label{sec:introductionCLT}\n\t\n\tThe asymptotics in Theorem~\\ref{thm:mainexpandingtranslates} affords the means to examine the long-term statistical behaviour of the averages of a given observable along expanding circle arcs. Historically, a momentous discovery in the twentieth century was the realization that the long-term evolution of deterministic systems  frequently obeys the same statistical laws governing the asymptotic behaviour of random processes. Specifically, this feature is a typical characteristic of dynamical systems with hyperbolic behaviour, among which geodesic flows on negatively curved compact manifolds feature prominently; we refer the reader to the survey in the introduction to~\\cite{Dolgopyat-Sarig}, as well as to the references therein, for an extensive discussion of the topic.  \n\t\n\tBecause of the exponential mixing properties of the geodesic flow $(\\phi^{X}_t)_{t\\in \\R}$ on $M$ (cf.~\\cite[Thm.~2]{Ratner}),  at first it stands to reason to expect, for a real-valued function $f$ on $M$ with finite first moment with respect to the volume measure $\\vol$, the distribution of the deviations from the average\n\t\\begin{equation*}\n\t\t\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_T^{X}\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\;,\n\t\\end{equation*}\n\twhen the base point $p$ is randomly chosen according to the law $\\vol$, (something which is henceforth indicated with $p\\sim \\vol$), to mimic for large values of $T$ the law of the empirical mean\n\t\\begin{equation*} \\frac{1}{N}\\sum_{n=1}^{N}X_n\n\t\\end{equation*}\n\tof an increasing number $N$ of independent real-valued random variables $X_n$. More precisely, since the hyperbolic length of a circle of radius $T$ is proportional to $e^{T}$ (see the explanation below~\\eqref{eq:integralbounds}), a full analogoue of the classical Central Limit Theorem in this case would affirm that the random variables \n\t\\begin{equation*}\n\t\te^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_{0}^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol \\biggr)\\;,\\quad p\\sim \\vol\n\t\\end{equation*} \n\tconverge in law, as $T$ tends to infinity, to a normally distributed random variable. However, the geometric resemblance of large hyperbolic circles to orbits of the unstable horocycle flow, for which similar phenomena occur (cf.~\\cite[Thm.~1.4,~1.5]{BuFo} and~\\cite[Thm.~4]{Rav}) accounts both for the emergence of other types of limiting distributions, and for possibly different renormalization factors, depending on the spectral properties of the observable under consideration.\n\t\n\tIn order to minimize the difference with the classical probabilistic setup of sums of independent random variables, we shall state all the results in this subsection for real-valued observables, though the extension to complex-valued ones is immediate.   \n\t\\begin{thm}\n\t\t\\label{thm:CLT}\n\t\tLet $\\theta\\in (0,4\\pi]$, $s>11\/2$, $f\\in W^{s}(M)$ a real-valued function. Assume that \n\t\t\\begin{equation*}\n\t\t\t\\mu_f=\\inf\\{\\mu\\in \\emph{Spec}(\\square)\\cap \\R_{>0}:D^{-}_{\\theta,\\mu}f \\emph{ does not vanish identically on }M \\}\n\t\t\\end{equation*}\n\t\tis finite, and let $\\nu_f$ be the unique complex number in $\\R_{\\geq 0}\\cup i\\R_{>0}$ satisfying $1-\\nu_f^{2}=4\\mu_f$.\n\t\t\\begin{enumerate}\n\t\t\t\\item If $0<\\mu_f<1\/4$, the random variables\n\t\t\t\\begin{equation*}\n\t\t\t\te^{\\frac{1-\\nu_f}{2}T}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\emph{d}s-\\int_{M}f\\;\\emph{d}\\vol\\biggr)\\;,\\quad p\\sim \\vol\n\t\t\t\\end{equation*}\n\t\t\tconverge in distribution to $D^{-}_{\\theta,\\mu_f}f$ as $T\\to\\infty$.\n\t\t\t\\item If $\\mu_f=1\/4$, the random variables\n\t\t\t\\begin{equation*}\n\t\t\t\tT^{-1}e^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\emph{d}s-\\int_{M}f\\;\\emph{d}\\vol\\biggr)\\;,\\quad p\\sim \\vol\n\t\t\t\\end{equation*}\n\t\t\tconverge in distribution to $D^{-}_{\\theta,1\/4}f$ as $T\\to\\infty$.\n\t\t\t\\item If $\\mu_f>1\/4$, the random variables\n\t\t\t\\begin{equation*}\n\t\t\t\te^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\emph{d}s-\\int_{M}f\\;\\emph{d}\\vol\\biggr)\\;,\\quad p\\sim \\vol\n\t\t\t\\end{equation*}\n\t\t\tconverge in distribution, as $T\\to\\infty$,  to the quasi-periodic motion\n\t\t\t\\begin{equation*} \\varepsilon_0D^{+}_{\\theta,1\/4}f(p)+\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f+\\sum_{\\mu \\in \\emph{Spec}(\\square),\\;\\mu\\geq \\mu_f}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{-}_{\\theta,\\mu}f\n\t\t\t\\end{equation*}\n\t\t\ton the set of real-valued random variables defined on the probability space $(M,\\vol)$.\n\t\t\\end{enumerate}\n\t\\end{thm}\n\t\n\tObserve the remarkable fact that the limiting distributions appearing in the statement of Theorem~\\ref{thm:CLT} are compactly supported on the real line, owing to the fact that the coefficients $D^{\\pm}_{\\theta,\\mu}f$ are bounded. This stands in stark contrast with the classical versions of the Central Limit Theorem in probability theory, where in non-trivial situations the distribution of errors is governed by the fully supported Gaussian distribution.\n\tFurthermore, it is straightforward to check, at least when the Casimir components of $f$ are eigenfunctions of $\\Theta$ and using the explicit expressions of the coefficients $D^{\\pm}_{\\theta,\\mu}f$ in~\\eqref{eq:Dplusabovequarter},~\\eqref{eq:Dminusabovequarter},~\\eqref{eq:Dquarter} and~\\eqref{eq:Dbelowquarter}, that the limit law is non-trivial\\footnote{As soon as $f$ is not almost-surely constant, obviously.}, that is, not a Dirac mass. In the general case, the coefficients are given by infinite superpositions of the previous ones; though we shall refrain from a detailed verification, there is no reason to expect cancellation phenomena to come about and produce limiting random variables which are constant almost-surely.\n\t\n\tTheorem~\\ref{thm:CLT} is a consequence of its quantitative version which we presently discuss. For $f$ fulfilling the conditions in Theorem~\\ref{thm:CLT}, let $\\mathbf{P}_{\\theta,f}$ denote the law of the random variable $D^{-}_{\\theta,\\mu_f}f$ when $\\mu_f\\leq 1\/4$, and $\\mathbf{P}_{\\theta,f}(T)$ the law of the random variable\n\t\\begin{equation*}\n\t\t\\varepsilon_0D^{+}_{\\theta,1\/4}f(p)+\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f+\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu\\geq \\mu_f}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{-}_{\\theta,\\mu}f\\;,\\quad T>0,\n\t\\end{equation*}\n\twhen $\\mu_f>1\/4$. Furthermore, for any $T\\geq 1$ we let $\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T)$ be:\n\t\\begin{enumerate} \n\t\t\\item the law of\n\t\t\\begin{equation*}\n\t\t\te^{\\frac{1-\\nu_f}{2}T}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\biggr) \\quad \\text{if }0<\\mu_f<1\/4,\n\t\t\\end{equation*}\n\t\t\\item the law of\n\t\t\\begin{equation*}\n\t\t\tT^{-1}e^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\biggr) \\quad \\text{if }\\mu_f=1\/4,\n\t\t\\end{equation*}\n\t\t\\item and the law of\n\t\t\\begin{equation*}\n\t\t\te^{\\frac{T}{2}}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_T\\circ r_s(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\biggr)\\quad \\text{if }\\mu_f>1\/4.\n\t\t\\end{equation*}\n\t\\end{enumerate}\n\t\n\tDenote by $d_{LP}$ the L\\'{e}vi-Prokhorov distance on the set of Borel probability measures on $\\R$ (cf.~Section~\\ref{sec:spatialDLT}), which induces on the latter the topology of weak convergence. Recall also that $\\mu_*$ denotes the spectral gap of $S=\\Gamma\\bsl \\Hyp$, with associated parameter $\\nu_*$.\n\t\n\n\t\\begin{prop}\n\t\t\\label{prop:CLT}\n\t\tLet the assumptions be as in Theorem~\\ref{thm:CLT}, and the constants $C_{\\emph{Spec}}$ and $C_{\\emph{Spec}}'$ be as in Theorem~\\ref{thm:mainexpandingtranslates}. \n\t\t\\begin{enumerate}\n\t\t\t\\item If $0<\\mu_f<1\/4$, then there is an explicit constant $\\eta_f>0$, depending only on $\\mu_f$ and on $\\emph{Spec}(\\square)$, such that \n\t\t\t\\begin{equation*}\n\t\t\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\emph{circ}}(T),\\mathbf{P}_{\\theta,f})\\leq \\frac{C'_{\\emph{Spec}}C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}}Te^{-\\eta_f T}\n\t\t\t\\end{equation*}\n\t\t\tfor every $T\\geq 1$.\n\t\t\t\\item If $\\mu_f=1\/4$, then there exists a constant $C_{\\emph{pos}}$, depending only on $\\emph{Spec}(\\square)\\cap \\R_{>0}$, such that \n\t\t\t\\begin{equation*}\n\t\t\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\emph{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq \\frac{C_{\\emph{pos}}C_{1,s-3} }{\\theta}\\norm{f}_{W^{s}}T^{-1} \n\t\t\t\\end{equation*}\n\t\t\tfor every $T\\geq 1$.\n\t\t\t\\item If $\\mu_f>1\/4$, then:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item when $\\mu_*<1\/4$,\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\emph{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq \\frac{C_{\\emph{Spec}}'C_{1,s-3}}{\\theta}\\norm{f}_{W^{s}}e^{-\\frac{\\nu_*}{2}T}\n\t\t\t\t\\end{equation*}\n\t\t\t\tfor every $T\\geq 1$;\n\t\t\t\t\\item when $\\mu_*\\geq 1\/4$,\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\emph{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq \\frac{ C_{\\emph{Spec}}C_{1,s-3} }{\\theta}\\norm{f}_{W^{s}}(T+1)e^{-\\frac{T}{2}}\n\t\t\t\t\\end{equation*}\n\t\t\t\tfor every $T\\geq 1$.\n\t\t\t\\end{enumerate}\n\t\t\\end{enumerate}\t\n\t\\end{prop}\n\t\n\tEntirely analogous deductions, of which we omit the details (cf.~Section~\\ref{sec:spatialDLT}), can be made in the case where the $D^{-}_{\\theta,\\mu}f$ vanish everywhere for any Casimir eigenvalue $\\mu>0$ but $D^{+}_{\\theta,\\mu}f$ is not identically zero for at least one such $\\mu$.\n\t\n\t\\begin{rmk}\n\t\tAs the proof of Proposition~\\ref{prop:CLT} shall clearly illustrate (see, in particular, Lemma~\\ref{lem:LPnearbyvariables}), the assumption that the base point $p$ is sampled according to the uniform measure $\\vol$ is immaterial, as far as the validity of Proposition~\\ref{prop:CLT} and Theorem~\\ref{thm:CLT} is concerned. It is possible to replace the measure $\\vol$ with any other Borel probability measure $\\mu$ on $M$, without affecting the quantitative rate of convergence, provided that the laws of the limiting random variables are modified accordingly. \n\t\\end{rmk}\n\t\n\tIn the remaining case when the $D^{\\pm}_{\\theta,\\mu}f$ vanish identically on $M$ for any positive Casimir eigenvalue $\\mu$, the explicit expressions of the coefficients appearing in Theorem~\\ref{thm:case_Theta_eigenfn} and~\\ref{thm:mainexpandingtranslates} enable us to rule out the existence of a non-trivial distributional limit in the case of full circles, that is, when $\\theta=4\\pi$. More precisely, we establish the following result.\n\t\n\t\\begin{thm}\n\t\t\\label{thm:noCLT}\n\t\tLet $s>11\/2$, $f\\in W^{s}(M)$ a real-valued function. Assume that, for any positive Casimir eigenvalue $\\mu$,  the functions $D^{\\pm}_{4\\pi,\\mu}f$ defined as in Theorem~\\ref{thm:mainexpandingtranslates} vanish identically on $M$. Then, for any collection $(B_T)_{T>0}$ of positive real numbers such that $B_T\\to\\infty$ as $T\\to\\infty$, the distributional limit of the random variables \n\t\t\\begin{equation*}\n\t\t\t\\frac{e^{T}\\bigl( \\frac{1}{4\\pi}\\int_0^{4\\pi}f\\circ \\phi_T^{X}\\circ r_s(p)\\;\\emph{d}s-\\int_{M}f\\;\\emph{d}\\vol\\bigr)}{B_T}\\;,\\quad p\\sim \\vol\n\t\t\\end{equation*} \n\t\tas $T\\to\\infty$ exists and is almost surely equal to zero.\n\t\\end{thm}\n\t\n\tTheorem~\\ref{thm:noCLT} results from approximating averages of $f$ along expanding circle arcs with the difference of the ergodic integrals of $f$ along two geodesic orbits, which in the case of complete circles happen to coincide; details are carried out in Section~\\ref{sec:noCLT}.\n\t\n\t\\begin{rmk}\n\t\t\\label{rmk:geodesiccoboundary}\n\t\tWe hasten to add that the argument we conduct enables to show that Theorem~\\ref{thm:noCLT} holds true for arbitrary circle arcs (that is, it is possible to replace $4\\pi$ with an arbitrary $\\theta\\in (0,4\\pi]$) provided that, in addition to the vanishing hypothesis on the $D^{\\pm}_{4\\pi,\\mu}$ for $\\mu\\in \\text{Spec}(\\square)\\cap \\R_{>0}$, the derivative $Uf_0$ along the stable horocycle flow of the projection $f_0$ of $f$ onto the Casimir eigenspace of eigenvalue $0$ is assumed to be a coboundary for the geodesic flow (cf.~Section~\\ref{sec:noCLT}). \n\t\\end{rmk}\n\t\n\t\n\t\n\t\\subsection{Asymptotics for arbitrary translates of compact orbits and the circle problem in the hyperbolic plane}\n\t\\label{sec:circleproblemhyperbolic}\n\t\n\tThe celebrated Gauss circle problem asks for the precise asymptotic behaviour of the discrepancy between the number of integer points in a disk of radius $R$ in the Euclidean plane and the area of the disk, as $R$ tends to infinity. More precisely, define the integer-point counting function \n\t\\begin{equation*}\n\t\t\\cN(R)=|\\{(m,n)\\in \\Z^{2}:m^{2}+n^2\\leq R^{2}  \\}|,\\quad R\\in \\R_{>0},\n\t\\end{equation*}\n\twhere $|A|$ denotes, here and henceforth, the cardinality of a finite set $A$.\n\tTessellating the Euclidean plane with $\\Z^{2}$-translates of $[0,1)^{2}$, which is a fundamental domain for the $\\Z^{2}$-action by translations on $\\R^{2}$, leads to the main term $\\pi R^{2}$, equal to the area of the disk of radius $R$, for the asymptotics of $\\cN(R)$, as well as to the upper bound (due to Gauss himself)\n\t\\begin{equation*}\n\t\t |\\cN(R)-\\pi R^{2}|\\leq 2\\pi(\\sqrt{2}R+1)\n\t\\end{equation*} \n\t\t for the discrepancy. Despite considerable successive improvements on Gauss' original bound, for the history of which we refer to the comprehensive survey~\\cite{Ivic}, it is a notoriously unsolved problem to attain the conjectural sharpest upper bound, deemed to be of the order of $R^{1\/2+\\varepsilon}$ for any $\\varepsilon>0$.\n\t\n\tWe consider the analogous question in the hyperbolic plane. For $\\Gamma<\\SL_2(\\R)$ a cocompact lattice, we examine the asymptotics of the function\n\t\\begin{equation}\n\t\t\\label{eq:defcountingfunction}\n\t\tN(R)=|\\{z\\in \\Gamma\\cdot i:d_{\\Hyp}(z,i)\\leq R\\}|\n\t\\end{equation}\n\tas $R$ tends to infinity, where $\\Gamma\\cdot i$ denotes the (discrete) orbit\\footnote{We thus count the number of actual lattice-orbit elements; we remark that, in the literature, the count of lattice elements $\\gamma\\in \\Gamma$ such that $d_{\\Hyp}(\\gamma\\cdot i,i)\\leq R$ often appears instead; the two quantities are proportional by a factor $|\\text{Stab}_{\\Gamma}(i)|$.} of $i$ under the $\\Gamma$-action on the hyperbolic plane $\\Hyp$ and we recall that $d_{\\Hyp}$ is the hyperbolic distance function on $\\Hyp$.\n\t\n\t\\begin{rmk}\n\t\tUpon replacing $\\Gamma$ by a conjugate, there is no loss of generality in choosing $i\\in \\Hyp$ as the base point: an elementary algebraic computation, together with the fact that $\\SL_2(\\R)$ acts by $d_{\\Hyp}$-isometries, leads to the equality\n\t\t\\begin{equation*}\n\t\t\t|\\{z\\in \\Gamma\\cdot w:d_{\\Hyp}(z,w)\\leq R  \\}|=|\\{z\\in g^{-1}\\Gamma g\\cdot i:d_{\\Hyp}(z,i)\\leq R\\}|\n\t\t\\end{equation*}\n\t\tfor any $w=g\\cdot i \\in \\Hyp$ and  $g\\in \\SL_2(\\R)$.\n\t\\end{rmk}\t\n\tThere is a compact fundamental domain for the action of $\\Gamma$ on $\\Hyp$, and a transposition of Gauss' tesselation argument to this setup provides a rationale for the heuristics concerning the main term of the asymptotics, which once again should be proportional to the hyperbolic area measure of the ball $B_R=\\{z\\in \\Hyp:d_{\\Hyp}(z,i)\\leq R \\}$, which we denote by $m_{\\Hyp}(B_R)$. However, a consequence of the peculiar features of hyperbolic geometry is that boundary effects become relevant, as opposed to the Euclidean setting: more precisely, the growth rate of the length of the boundary $\\partial{B_R}$ turns out to be equal to the growth rate of $m_{\\Hyp}(B_R)$. The error rate resulting from the tesselation approach is consequently of the same order of the main term, and as such meaningless.\n\t\n\tAs for its Euclidean counterpart, the circle problem in the hyperbolic plane has been the subject of intensive research over the course of the twentieth century, with fundamental contributions due to Delsarte (\\cite{Delsarte}), Selberg (\\cite{Selberg}), Margulis (\\cite{Margulis}), Lax and Phillips (\\cite{Lax-Phillips}) and Phillips and Rudnick (\\cite{Phillips-Rudnick}, see its introduction for a detailed history of the problem). To a large extent, the state of the art concerning the best estimate on the error term $|N(R)-c_{\\Gamma}m_{\\Hyp}(B_R)|$,\n\twhere $c_{\\Gamma}$ is an explicit constant which we identify in Theorem~\\ref{thm:countingproblem}, \n\tis represented by Selberg's upper bound\n\t\\begin{equation}\n\t\t\\label{eq:Selbergbound} |N(R)-c_{\\Gamma}m_{\\Hyp}(B_R)|\\leq e^{(\\sup\\{2\/3,(1+\\Re{\\nu_*})\/2\\})R}\\;,\n\t\\end{equation}\n\twhere recall that $(1-\\nu_*^{2})\/4$ is the spectral gap of $S=\\Gamma\\bsl \\Hyp$. The estimate\\footnote{As a matter of fact, Selberg's asymptotics is more accurate than the one recorded here, featuring a main term which involves, beside the hyperbolic area of the balls, additional terms depending on Laplace eigenfunctions for small eigenvalues; see~\\cite[Eq.~1.13]{Phillips-Rudnick}.} in~\\eqref{eq:Selbergbound} (which is equally valid for non-uniform lattices $\\Gamma<\\SL_2(\\R)$) is obtained by means of a deep analysis of a  class of integral operators commuting with hyperbolic isometries (cf.~\\cite{Selberg}).\n\t\n\tIt was Margulis' realization (see~\\cite{Margulis-thesis}) that lattice point counting problems of the type we are examining are intimately interwoven with questions of equidistribution of translates of subgroup orbits on homogeneous spaces. Subsequently, this novel perspective was profitably pursued and vastly generalized in the works of Duke, Rudnick and Sarnak~\\cite{Duke-Rudnick-Sarnak} and Eskin and McMullen~\\cite{Eskin-McMullen}. Specializing to our current setup, it turns out that the distribution properties of translates of $\\SO_2(\\R)$-orbits\\footnote{We consider here the canonical left action $g\\cdot \\Gamma g'=\\Gamma g'g^{-1}$ of $\\SL_2(\\R)$ on $M$.} on $M= \\Gamma\\bsl \\SL_2(\\R)$ lead to meaningful information on the growth rate of $N(R)$, in a way that is amenable to quantitative refinements (see Section~\\ref{sec:latticepoint} for an extensive treatment of the connection in its quantitative form).    \n\t\n\tWe are thus lead to study effective equidistribution properties of $\\SO_2(\\R)$-orbits on $M$, which can be readily derived from Theorem~\\ref{thm:mainexpandingtranslates} via the standard Cartan decomposition for $\\SL_2(\\R)$.  For any $g_0\\in \\SL_2(\\R)$, define the right-translation map $R_{g_0}\\colon M\\to M$ by $R_{g_0}(\\Gamma g)=\\Gamma g g_0$ for any $g\\in \\SL_2(\\R)$. For every $p \\in M$, we indicate with $m_{\\SO_2(\\R)\\cdot p}$ the unique $\\SO_2(\\R)$-invariant Borel probability measure on $M$ which is fully supported on the (compact) $\\SO_2(\\R)$-orbit of the point $p$; furthermore, for any $g\\in \\SL_2(\\R)$, the notation $g_*m_{\\SO_2(\\R)\\cdot p}$ stands for the push-forward of $m_{\\SO_2(\\R)\\cdot p}$ under the action of $g$, which clearly depends only on the left coset of $g$ modulo $\\SO_2(\\R)$. Making use of the mixing properties of the global $\\SL_2(\\R)$-action on $M$ via a clever thickening argument, Margulis proved  (see~\\cite{Margulis,Margulis-thesis}) that arbitrary translates of $m_{\\SO_2(\\R)\\cdot p}$ equidistribute towards the $\\SL_2(\\R)$-invariant measure $\\vol$; this amounts to the fact that, for any continuous function $f\\colon M\\to \\C$, \n\t\\begin{equation*}\n\t\t\\int_{M}f\\;\\text{d}g_*m_{\\SO_2(\\R)\\cdot p}\\longrightarrow\\int_{M}\\;f\\;\\text{d}\\vol\n\t\\end{equation*} \n\tas $g\\SO_2(\\R)$ tends to infinity in the quotient $\\SL_2(\\R)\/\\SO_2(\\R)$.\n\t\n\tIn order to phrase a quantitative version of the previous statement conveniently, we introduce the notation $\\norm{g}_{\\text{op}}$ to indicate operator norm of an element $g\\in \\SL_2(\\R)$ with respect to the standard Euclidean norm on $\\R^{2}$; such a specific choice, while obviously immaterial, arises naturally over the course of the proof. \n\t\n\n\t\n\t\\begin{thm}\n\t\t\\label{thm:mainarbitrarytranslates}\n\t\tLet $C_{\\emph{Spec}},C'_{\\emph{Spec}}$ be as in Theorem~\\ref{thm:mainexpandingtranslates}, $s>11\/2$ a real number. There exists a real constant $C_{s,\\emph{rot}}$, depending only on $s$ and on $M$, such that the following holds:\n\t\tif $f\\in W^{s}(M)$, then there exist, for any positive Casimir eigenvalue $\\mu$, functions $D^{+}_{\\mu}f,\\;D^{-}_{\\mu}f\\colon M\\times \\SL_2(\\R)\\to \\C$ with\n\t\t\\begin{equation*}\n\t\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)\\cap \\R_{>0}}\\sup\\limits_{p \\in M,\\;g\\in \\SL_2(\\R)}|D^{\\pm}_{\\mu}f(p,g)|\\leq C_{s,\\emph{rot}}C_{\\emph{Spec}}' C_{1,s-3}\\norm{f}_{W^{s}}\\;,\n\t\t\\end{equation*}\n\t\tsuch that, for every $p \\in M$ and every $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\emph{op}}\\geq \\sqrt{e}$,\n\t\t\\begin{equation*}\n\t\t\t\\begin{split}\n\t\t\t\t\\int_{M}f\\;\\emph{d}&g_{*}m_{\\SO_2(\\R)\\cdot p}=\\int_{M}f\\;\\emph{d}\\vol\\\\\n\t\t\t\t&+\\norm{g}_{\\emph{op}}^{-1}\\biggl(\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;\\mu>1\/4}\\cos{(\\Im{\\nu}\\log{\\norm{g}_{\\emph{op}}})}D^{+}_{\\mu}f(p,g)+\\sin{(\\Im{\\nu}\\log{\\norm{g}_{\\emph{op}}})}D^{-}_{\\mu}f(p,g)\\biggr)\\\\\n\t\t\t\t&+\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;0<\\mu<1\/4}\\norm{g}_{\\emph{op}}^{-(1+\\nu)}D^{+}_{\\mu}f(p,g)+\\norm{g}_{\\emph{op}}^{-(1-\\nu)}D^{-}_{\\mu}f(p,g)\\\\\n\t\t\t\t&+\\varepsilon_0\\bigl(\\norm{g}_{\\emph{op}}^{-1}D^{+}_{1\/4}f(p,g)+2\\norm{g}_{\\emph{op}}^{-1}\\log{\\norm{g}_{\\emph{op}}}D^{-}_{1\/4}f(p,g)\\bigr)+\\mathcal{R}f(p,g)\\;,\n\t\t\t\\end{split}\n\t\t\\end{equation*}\n\t\twhere \n\t\t\\begin{equation*}\n\t\t\t|\\mathcal{R}f(p,g)|\\leq C_{\\emph{Spec}}C_{1,s-3}C_{s,\\emph{rot}} \\norm{f}_{W^{s}}(2\\log{\\norm{g}_{\\emph{op}}}+1)\\norm{g}_{\\emph{op}}^{-2}\\;.\n\t\t\\end{equation*}\n\t\\end{thm} \n\t\n\t\\begin{rmk}\n\t\t\\begin{enumerate}\n\t\t\t\\item The problem of effective equidistribution of translates of finite-volume orbits, in the vastly more general context of affine symmetric spaces, was thoroughly explored by Benoist and Oh in~\\cite{Benoist-Oh}. Their approach relies crucially on effective bounds for the mixing rates\\footnote{Specializing to our setup, effective mixing for the $\\SL_2(\\R)$-action on finite-volume quotients $\\Gamma\\bsl \\SL_2(\\R)$ was first worked out in detail, to the best of our knowledge, by Kleinbock and Margulis in~\\cite[Sec.~2.4]{Kleinbock-Margulis} (see also \\cite{Ratner}).} of the relevant global action, and systematically developes quantitative versions of the geometric properties which play a major role in the original work~\\cite{Eskin-McMullen} of Eskin and McMullen.  In the specific instance of the affine symmetric space being the hyperbolic plane $\\Hyp$,  Theorem~\\ref{thm:mainarbitrarytranslates} improves upon~\\cite[Thm.~1.10]{Benoist-Oh} in that it quantifies the exponent governing the equidistribution rate and spells out additional terms in the asymptotic expansion.\n\t\t\t\\item Just as in the case of Theorem~\\ref{thm:mainexpandingtranslates} and Corollary~\\ref{cor:effective}, the asymptotic expansion in Theorem~\\ref{thm:mainarbitrarytranslates} delivers at once the (optimal) effective equidistribution bound\n\t\t\t\\begin{equation*}\n\t\t\t\t\\biggl|\\int_{M}f\\;\\text{d}g_*m_{\\SO_2(\\R)\\cdot p}-\\int_{M}f\\;\\text{d}\\vol\\biggr|\\leq D^{\\text{main}}f(p,g)\\;(\\log{\\norm{g}_{\\text{op}}})^{\\varepsilon_0}\\norm{g}_{\\text{op}}^{-(1-\\Re{\\nu_*})}\n\t\t\t\\end{equation*}\n\t\t\tfor every $p \\in M$ and $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\text{op}}\\geq \\sqrt{e}$, \n\t\t\twhere the function $D^{\\text{main}}\\colon M\\times \\SL_2(\\R)\\to \\C$ is uniformly bounded in terms of an appropriate Sobolev norm of $f$ and of spectral data depending only on $M$.  Following the thread of the observations expressed in Remark~\\ref{rmk:equivalentmetrics}, it is instructive to  compare it with the decay rates for matrix coefficients of unitary representations of $\\SL_2(\\R)$ computed by Venkatesh in~\\cite[Sec.~9.1.2]{Venkatesh}.\n\t\t\\end{enumerate}\n\t\\end{rmk}\n\t\n\tTheorem~\\ref{thm:mainarbitrarytranslates} affords a precise asymptotic formula for the averaged counting of points on translates of $\\Gamma$-orbits inside balls of increasing radius, that is, for quantities of the form\\footnote{As it is customary in the literature addressing such themes, we shall choose to work with spaces of left cosets whenever dealing with the lattice point counting problem. The map $g\\Gamma\\mapsto \\Gamma g^{-1}$ establishes an $\\SL_2(\\R)$-equivariant diffeomorphism, between $\\SL_2(\\R)\/\\Gamma$ and $\\Gamma\\bsl \\SL_2(\\R)$; every object we have defined on $\\Gamma\\bsl \\SL_2(\\R)$ shall thus be identified (without altering notation) with the corresponding object in $\\SL_2(\\R)\/\\Gamma$ without further comment.}\n\t\\begin{equation}\n\t\t\\label{eq:averagecounting}\n\t\t\\int_{\\SL_2(\\R)\/\\Gamma}\\frac{|g\\Gamma\\cdot i\\cap B_R|}{m_{\\Hyp}(B_R)}\\psi(g\\Gamma )\\;\\text{d}\\vol(g\\Gamma)\n\t\\end{equation}\n\tas $\\norm{g}_{\\text{op}}$ tends to infinity, where $\\psi$ is a sufficiently regular function on $\\SL_2(\\R)\/\\Gamma$. Prior to the statement of the result, we shall fix advantageous normalizations for the various invariant measures involved (see Section~\\ref{sec:averagecounting} for the details).\n\t\n\tLet $m_{\\SL_2(\\R)}$ be the unique choice of Haar measure on $\\SL_2(\\R)$ such that, if $m_{\\SO_2(\\R)}$ is the probability Haar measure on the compact subgroup $\\SO_2(\\R)$, then $m_{\\SL_2(\\R)}$ is the (formal) product (cf.~Proposition~\\ref{prop:foldingunfolding}) of $m_{\\SO_2(\\R)}$ and of the  $\\SL_2(\\R)$-invariant measure on the homogeneous space $\\SL_2(\\R)\/\\SO_2(\\R)$ which corresponds, under the canonical identification of the latter space with $\\Hyp$, to the hyperbolic area measure $m_{\\Hyp}$. We then indicate with $m_{\\SL_2(\\R)\/\\Gamma}$ the unique $\\SL_2(\\R)$-invariant measure on $\\SL_2(\\R)\/\\Gamma$ such that $m_{\\SL_2(\\R)}$ is the product of $m_{\\SL_2(\\R)\/\\Gamma}$ and the counting measure on the discrete group $\\Gamma$. Observe that $m_{\\SL_2(\\R)\/\\Gamma}$ is a scalar multiple of the probability measure $\\vol$, the multiplying factor being equal to the covolume\n\t\\begin{equation*} \\text{covol}_{\\SL_2(\\R)}(\\Gamma)=m_{\\SL_2(\\R)\/\\Gamma}(\\SL_2(\\R)\/\\Gamma)\n\t\\end{equation*}\n\tof the lattice $\\Gamma$ inside $\\SL_2(\\R)$. We denote similarly by $\\text{covol}_{\\SO_2(\\R)}(\\Gamma\\cap \\SO_2(\\R))$ the volume\\footnote{A straightfoward application of the formula in~\\eqref{eq:foldingunfolding} shows that this equals the reciprocal of the cardinality of the finite group $\\Gamma\\cap \\SO_2(\\R)$.} of the compact quotient $\\SO_2(\\R)\/(\\Gamma\\cap \\SO_2(\\R))$ with respect to the $\\SO_2(\\R)$-invariant measure induced by $m_{\\SO_2(\\R)}$ and the counting measure on $\\Gamma\\cap \\SO_2(\\R)$.\n\t\n\tLastly, recall that $B_R$ is the closed hyperbolic ball in $\\Hyp$ of radius $R>0$ centered at $i$.\n\t\n\t\\begin{prop}\n\t\t\\label{prop:averagedcounting}\n\t\tLet $C_{\\emph{Spec}},C_{\\emph{Spec}}'$ be as in Theorem~\\ref{thm:mainexpandingtranslates}. Suppose given a real number $s>11\/2$ and a function $\\psi\\in W^{s}(\\SL_2(\\R)\/\\Gamma)$. There exist, for any positive Casimir eigenvalue $\\mu$, functions $\\beta_{\\psi,\\mu}^{+},\\beta_{\\psi,\\mu}^{-}\\colon \\R_{>0}\\to \\C$  with \n\t\t\\begin{equation}\n\t\t\t\\label{eq:betabound}\n\t\t\t\\sum_{\\mu\\in \\emph{Spec}(\\square)\\cap \\R_{>0}}\\sup_{R>0}|\\beta_{\\psi,\\mu}^{\\pm}(R)|\\leq \\frac{C_{\\emph{Spec}}'C_{1,s-3} }{2\\pi}\\norm{\\psi}_{W^{s}}\n\t\t\\end{equation}\n\t\tsuch that, for every $R\\geq 1$, \n\t\t\\begin{equation}\n\t\t\t\\label{eq:averagedcountingfunction}\n\t\t\t\\begin{split}\n\t\t\t\t\\frac{1}{\\emph{covol}_{\\SO_2(\\R)}(\\Gamma\\cap \\SO_2(\\R))}&\\int_{\\SL_2(\\R)\/\\Gamma} \\frac{|g\\Gamma\\cdot i \\cap B_R |}{m_{\\Hyp}(B_R)}\\;\\psi(g\\Gamma )\\;\\emph{d}m_{\\SL_2(\\R)\/\\Gamma}(g\\Gamma)=\\\\\n\t\t\t\t&\\frac{1}{\\emph{covol}_{\\SL_2(\\R)}(\\Gamma)}\\int_{\\SL_2(\\R)\/\\Gamma}\\psi\\;\\emph{d}m_{\\SL_2(\\R)\/\\Gamma}\\\\\n\t\t\t\t&\n\t\t\t\t+ e^{-\\frac{R}{2}}\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;\\mu>1\/4}\\beta_{\\psi,\\mu}^{+}(R)+\\beta_{\\psi,\\mu}^{-}(R)\\\\\n\t\t\t\t&\n\t\t\t\t+\\sum_{\\mu\\in \\emph{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}R}\\beta^{+}_{\\psi,\\mu}(R)+e^{-\\frac{1-\\nu}{2}R}\\beta^{-}_{\\psi,\\mu}(R)\\\\\n\t\t\t\t& +\\varepsilon_0\\biggl(e^{-\\frac{R}{2}}\\beta^{+}_{\\psi,1\/4}(R)+Re^{-\\frac{R}{2}}\\beta^{-}_{\\psi,1\/4}(R)\\biggr)+\\gamma_{\\psi}(R)\\;,\n\t\t\t\\end{split}\n\t\t\\end{equation}\n\t\twhere \n\t\t\\begin{equation}\n\t\t\t\\label{eq:gammabound}\n\t\t\t|\\gamma_{\\psi}(R)|\\leq \\frac{5C_{\\emph{Spec}}C_{1,s-3}}{4\\pi}\\norm{\\psi}_{W^{s}} (R+1)e^{-R}\\quad .\n\t\t\\end{equation}\n\t\\end{prop}\n\t\n\tRecall now our definition of the counting function $N(R)$ in~\\eqref{eq:defcountingfunction}. An optimization argument involving approximate identities on the homogeneous space $\\SL_2(\\R)\/\\Gamma$ enables us to derive information on the asymptotic behaviour of the error term in the pointwise counting problem discussed at the beginning of this subsection. We remind the reader that $\\nu_*$ is the unique complex number in $\\R_{\\geq 0}\\cup i\\R_{>0}$ such that $\\frac{1-\\nu_*^{2}}{4}$ equals the spectral gap of $S=\\Gamma\\bsl \\Hyp$.\n\t\n\t\\begin{thm}\n\t\t\\label{thm:countingproblem}\n\t\tLet $\\Sigma\\colon \\R_{>0}\\to \\R_{>0}$ be the function defined by\n\t\t\\begin{equation*}\n\t\t\t\\Sigma(R)=\\frac{\\emph{covol}_{\\SO_2(\\R)}(\\Gamma\\cap \\SO_2(\\R))}{\\emph{covol}_{\\SL_2(\\R)}(\\Gamma)}m_{\\mathbb{H}}(B_R)\\;, \\quad R>0.\n\t\t\\end{equation*}\n\t\tSet $\\eta_*=\\frac{1}{13}(1-\\Re{\\nu_*})$. Then,\n\t\tfor every $\\varepsilon>0$, \n\t\t\\begin{equation*}\n\t\t\tN(R)=\\Sigma(R)+o(e^{(1-\\eta_*+\\varepsilon)R})\n\t\t\\end{equation*}\n\t\tas $R$ tends to infinity.\n\t\\end{thm}\n\t\n\t\\begin{rmk}\n\t\tAs it will emerge in the proof of Theorem~\\ref{thm:countingproblem}, which is detailed in Section~\\ref{sec:countingproblem}, the appearance of the quantity $1\/13$ in the exponent is ultimately an outgrowth of the minimal Sobolev regularity of the test function $f$ we need to impose in Theorem~\\ref{thm:expandingonsurface}, which in turn is needed because of the upper bounds in~\\eqref{eq:boundDthetamu} and~\\eqref{eq:globalremainderestimate} depending on the Sobolev norm $\\norm{f}_{W^s}$ for some $s>9\/2$. In this regard, we observe the following: suppose that, in the latter two bounds, the norm $\\norm{f}_{W^{s}}$ can be replaced by $\\norm{f}_{W^1}$, as it is the case for joint eigenfunctions of $\\square$ and $\\Theta$ (cf.~Theorem~\\ref{thm:case_Theta_eigenfn}); then Theorem~\\ref{thm:countingproblem} would hold with $1\/6$ in place of $1\/13$. \n\t\\end{rmk}\n\t\n\t\\subsection{Outline of the proofs and layout of the article}\n\t\n\tThe method we employ to prove Theorem \\ref{thm:case_Theta_eigenfn} was originally devised by Ratner in~\\cite{Ratner}, who realized that the problem of finding mixing rates for geodesic and horocycle flows can be reduced to solving a family of linear second-order ordinary differential equations\\footnote{After the completion of a first draft of the present article, the authors were made aware of the unpublished manuscript~\\cite{Edwards-unpublished} by S.~Edwards, in which a weaker formulation of the quantitative equidistribution result in Theorem~\\ref{thm:mainexpandingtranslates} is provided. The strategy of proof is entirely analogous to the one pursued here, and is there applied, more generally, to the quantitative investigation of the equidistribution properties of translated orbits of symmetric subgroups on homogeneous spaces of semisimple Lie groups.}. \n\tThis ingenuous and yet fairly elementary approach has been further developed by Burger in~\\cite{Bur} to prove polynomial bounds for the equidistribution of horocycle orbits in compact quotients of $\\SL_2(\\R)$. Later, Str\\\"{o}mbergsson \\cite{Str} and Edwards \\cite{Edw} exploited the same idea to study effective equidistribution properties of unipotent orbits in more general settings.\n\tIn the same spirit, the second author recently provided (see~\\cite{Rav}), using Ratner's strategy, an alternative proof of Flaminio-Forni's asymptotic expansion for horocycle ergodic integrals \\cite{Flaminio-Forni}. \n\t\n\tWe begin in Section~\\ref{sec:preliminaries} with an overview of the required notions concerning hyperbolic surfaces, Sobolev spaces and harmonic analysis on the Lie group $\\SL_2(\\R)$, nailing down notation to be employed throughout the manuscript.\n\tAssuming that a $\\cC^{2}$-observable $f$ is a joint eigenfunction of the Casimir operator and of the vector field $\\Theta$, we then show in Section~\\ref{sec:reductiontoODE} that the behaviour of the circle-arc averages $k_{f, \\theta}(p, \\cdot)$ (cf.~\\eqref{eq:ktheta}), viewed as functions of the time $t$ for a fixed base point $p \\in \\Gamma\\bsl \\SL_2(\\R)$, fulfill a second order linear ODE, solving which explicitly leads to the proof of Theorem~\\ref{thm:case_Theta_eigenfn} presented in Section~\\ref{sec:jointeigenfn}; incidentally, we may arrange computations so that the latter takes on the same form of the ODE satisfied by time rescalings of horocycle averages in~\\cite{Rav} (see, in particular,~\\cite[Prop.~8]{Rav}), which accounts for the similarity between Theorem~\\ref{thm:case_Theta_eigenfn} with~\\cite[Thm.~1]{Rav}.\n\tIn Section~\\ref{sec:arbitraryfn}, the asymptotic expansion of Theorem \\ref{thm:mainexpandingtranslates} is deduced from Theorem \\ref{thm:case_Theta_eigenfn} taking advantage of a few elementary facts from the classical harmonic analysis of $\\SL_2(\\R)$. Additional regularity on $f$ is required in order to ensure the absolute convergence of the expansion in \\eqref{eq:asymptoticgeneral}; see, in particular, Section~\\ref{sec:sumestimates}. The asymptotics for arbitrary translates in Theorem~\\ref{thm:mainarbitrarytranslates} is derived from Theorem~\\ref{thm:mainexpandingtranslates} in Section~\\ref{sec:arbitrarytranslates}.\n\tBuilding on Theorem~\\ref{thm:mainexpandingtranslates} once more, we establish in Sections~\\ref{sec:spatialDLT} and~\\ref{sec:noCLT} the distributional limit Theorems~\\ref{thm:CLT} and~\\ref{thm:noCLT} for the random variable $k_{f,\\theta}(p,t)$, appropriately centered and normalized, when the initial point $p$ is taken randomly with respect to the uniform measure.\n\tThese limit theorems mirror those for horocycle ergodic integrals, for which we refer the reader to~\\cite{BuFo, Rav}. Section~\\ref{sec:temporalDLT} hosts a few considerations concerning the point of view of temporal distributional limit theorems (see~\\cite{Dolgopyat-Sarig}) on the problem of analyzing the statistical behaviour of circle averages. \n\tFinally, in Section~\\ref{sec:latticepoint}, we provide a quantitative treatment of the approach of Duke-Rudnick-Sarnak~\\cite{Duke-Rudnick-Sarnak} and Eskin-McMullen~\\cite{Eskin-McMullen}, which allows to prove both Proposition~\\ref{prop:averagedcounting} and Theorem~\\ref{thm:countingproblem} on the hyperbolic lattice point counting problem.\n\t\n\t\n\t\n\t\n\t\\section{Preliminaries on harmonic analysis on $\\SL_2(\\R)$}\n\t\\label{sec:preliminaries}\n\tIt is the aim of this section to carefully describe the setting of our main results as well as to review the required notions on the representation theory of the group $\\SL_2(\\R)$ which will play a central role throughout the article. \n\t\n\t\n\t\\subsection{Hyperbolic surfaces and their unit tangent bundles}\n\t\\label{sec:hyperbolic}\n\tThe subject of this subsection is classical: detailed treatments can be found, for instance, in~\\cite{Bekka-Mayer,Bergeron,Borel,Buser,Einsiedler-Ward,Iwaniec,Katok}.\n\t\n\tThe special linear group $\\SL_2(\\R)$ is the group of $2\\times 2$ real matrices with determinant $1$. It is a three-dimensional Lie group, whose Lie algebra we denote by $\\sl_2(\\R)$ and identify canonically with the Lie algebra of traceless $2\\times 2$  matrices with real entries. The identity matrix in $\\SL_2(\\R)$ is denoted by $I_2$. A basis of $\\sl_2(\\R)$ as a real vector space is given by the elements\n\t\\begin{equation*}\n\t\tX=\n\t\t\\begin{pmatrix}\n\t\t\t1\/2&0\\\\0&-1\/2\n\t\t\\end{pmatrix}\n\t\t, \\quad U=\n\t\t\\begin{pmatrix}\n\t\t\t0&1\\\\\n\t\t\t0&0\n\t\t\\end{pmatrix}\n\t\t, \\quad V=\n\t\t\\begin{pmatrix}\n\t\t\t0&0\\\\\n\t\t\t1&0\n\t\t\\end{pmatrix}\n\t\t.\n\t\\end{equation*}\n\tWith $\\exp\\colon \\sl_2(\\R)\\to \\SL_2(\\R)$ we indicate the exponential map, and with $\\Ad\\colon \\SL_2(\\R)\\to \\GL(\\sl_2(\\R))$, $g\\mapsto \\Ad_g$ the adjoint representation of $\\SL_2(\\R)$ onto $\\sl_2(\\R)$.\n\t\n\t\n\tLet $\\Hyp=\\{z=x+iy\\in \\C:y>0 \\}$ be the Poincar\\'{e} upper-half plane, endowed with the Riemannian metric\n\t\\begin{equation*}\n\t\tg_{(x,y)}=\\frac{(\\text{d}x)^{2}+(\\text{d}y)^{2}}{y^2}\\;, \\quad (x,y)\\in \\Hyp.\n\t\\end{equation*} \n\tThe Riemannian manifold $(\\Hyp,g)$ is a model of the hyperbolic plane, that is, of the unique complete simply connected two-dimensional Riemannian manifold of constant sectional curvature equal to $-1$ (cf.~\\cite[Part 1 Chap.~6]{Bridson-Haefliger}). \n\tThe Lie group $\\SL_2(\\R)$ acts smoothly by orientation-preserving\\footnote{Actually, the action is by analytic transformations of the Riemann surface $\\Hyp$.} isometries of the hyperbolic plane. The action is given by the M\\\"{o}bius transformations\n\t\\begin{equation*}\n\t\t\\begin{pmatrix}\n\t\t\ta&b\\\\\n\t\t\tc&d\n\t\t\\end{pmatrix}\n\t\t\\cdot z=\\frac{az+b}{cz+d}\\;,\\quad a,b,c,d\\in \\R, \\;ad-bc=1,\\;z\\in \\Hyp;\n\t\\end{equation*}\n\tit is transitive, and thus gives rise to an $\\SL_2(\\R)$-equivariant diffeomorphism between $\\Hyp$ and the quotient manifold $\\SL_2(\\R)\/\\SO_2(\\R)$, where the special orthogonal group $\\SO_2(\\R)$ is the stabilizer of the point $i\\in \\Hyp$.\n\t\n\tLet $\\Gamma<\\SL_2(\\R)$ be a cocompact lattice\\footnote{We recall that a lattice in a locally compact Hausdorff topological group $G$ is a discrete subgroup $\\La<G$ such that the quotient space $\\La\\bsl G$ admits a non-zero $G$-invariant Radon probability measure. The lattice $\\La$ is cocompact if $\\La\\bsl G$ is a compact topological space.}. If the projection of $\\Gamma$ to the projective special linear group $\\PSL_2(\\R)=\\SL_2(\\R)\/\\{\\pm I_2 \\}$ has no non-trivial torsion elements, then the quotient space $S=\\Gamma\\bsl \\Hyp$, homeomorphic to the double coset space $ \\Gamma\\bsl \\SL_2(\\R)\/\\SO_2(\\R)$, is a compact, connected, orientable smooth surface inheriting from $\\Hyp$ a Riemannian metric $g_{S}$ of constant sectional curvature equal to $-1$, that is, a hyperbolic Riemannian metric.  Conversely, it is a well-known consequence of Poincar\\'{e}-Koebe's Uniformization Theorem (cf.~\\cite[Chap.~IV]{Farkas-Kra}) that any compact, connected, orientable hyperbolic surface is isomorphic, as a Riemannian manifold, to a quotient $\\Gamma\\bsl\\Hyp$, with $\\Gamma<\\SL_2(\\R)$ a cocompact lattice with torsion-free projection on $\\PSL_2(\\R)$. In case the projection of $\\Gamma$ to $\\PSL_2(\\R)$ has non-trivial torsion elements, the quotient $S=\\Gamma\\bsl \\Hyp$ is, more generally, a hyperbolic \\emph{orbifold} (see~\\cite[Chap.~13]{Ratcliffe} and~\\cite[Chap.~13]{Thurston}). Unifying, though mildly abusing terminology, we shall throughout refer to $S$ as a surface in both the previous cases. \n\t\n\tThe simply transitive action of $\\PSL_2(\\R)$ on the unit tangent bundle of $\\Hyp$ allows to identify, as smooth manifolds, the unit tangent bundle $T^{1}S=\\{(p,v)\\in TS:\\norm{v}_{g_S}=1 \\}$ of $S$ with the homogeneous space $M=\\Gamma\\bsl \\SL_2(\\R)$. Throughout this manuscript, we shall solely appeal to the algebraic structure of $M$ as a homogeneous space of $\\SL_2(\\R)$, and not to its geometric nature of principal circle bundle over the surface $S$.\n\t\n\tFor any $r\\in \\N\\cup\\{\\infty\\}$, we denote by $\\mathscr{C}^r(M)$ the vector space of complex-valued functions of class $\\mathscr{C}^{r}$ defined on $M$. It is endowed with the norm $\\norm{\\cdot}_{\\cC^{r}}$ defined in~\\eqref{eq:Cknorm}. The $\\mathscr{C}^{0}$-norm is abbreviated with $\\norm{\\cdot}_{\\infty}$.\n\t\n\tA vector $W\\in \\sl_2(\\R)$ gives rise to the one-parameter subgroup $(\\exp(tW))_{t\\in \\R}$ of $\\SL_2(\\R)$, which in turn acts as a smooth flow on $M$ by right translations. Whenever convenient, we identify $W$ with the infinitesimal generator of such a flow, and denote its action as a derivation on the function space $\\mathscr{C}^{1}(M)$ by $f\\mapsto Wf$. This extends to an isomorphism of associative $\\R$-algebras between the universal enveloping algebra $U(\\sl_2(\\R))$ of $\\sl_2(\\R)$ and the algebra of $\\SL_2(\\R)$-invariant differential operators on $\\mathscr{C}^{\\infty}(M)$. More generally, any element $Y\\in U(\\sl_2(\\R))$ of order $k\\in \\N$ acts naturally on $\\mathscr{C}^{k}(M)$, the action being also denoted by $f\\mapsto Yf$. Observe that, for any $k\\in \\N$, the definition of the $\\mathscr{C}^k$-norm on $\\mathscr{C}^k(M)$ implies that $\\norm{Yf}_{\\infty}\\leq \\norm{f}_{\\mathscr{C}^{k}}$ for any $Y\\in U(\\sl_2(\\R))$ of order at most $k$ and any $f\\in \\mathscr{C}^{k}(M)$.\n\t\n\t\\subsection{Unitary representations, the Casimir and Laplace operators, Weyl's law}\n\t\\label{sec:unitaryrepresentations}\n\t\n\tConvenient sources for the material presented in this subsection, in addition to those cited in Section~\\ref{sec:hyperbolic}, are~\\cite{Lang, Mackey,Sarnak, Terras}.\n\t\n\tLet us denote by $H$ be the complex Hilbert space $L^{2}(M)$ of functions which are square-integrable with respect to the $\\SL_2(\\R)$-invariant measure $\\vol$ on $M$; the inner product defining the Hilbert space structure on $H$ shall be denoted by $\\langle \\cdot, \\cdot \\rangle$. With $\\mathscr{U}(H)$ we indicate the group of unitary operators $T\\colon H\\to H$.\n\tThe standard smooth (left) action of $\\SL_2(\\R)$ on the homogeneous space $M$, given by $g_0\\cdot \\Gamma g=\\Gamma gg_0^{-1}$ for every $g_0,g\\in \\SL_2(\\R)$, preserves the measure $\\vol$, and hence gives rise to a unitary representation $\\rho\\colon \\SL_2(\\R)\\to \\mathscr{U}(H),\\;g\\mapsto \\rho_g$, called the quasi-regular representation of $\\SL_2(\\R)$ on $H$. \n\t\n\tFor any $r\\in \\N\\cup\\{\\infty\\}$, denote by $\\mathscr{C}^{r}(H)$ the linear subspace of $\\mathscr{C}^{r}$-vectors on $H$, that is,\n\t\\begin{equation*}\n\t\t\\mathscr{C}^{r}(H)=\\{v\\in H: \\text{ the map }\\SL_2(\\R)\\ni g\\mapsto \\rho_g(v)\\in H \\text{ is of class }\\mathscr{C}^{r}  \\}.\n\t\\end{equation*}\n\t\n\tAn element $W\\in \\sl_2(\\R)$ induces a linear operator $L_{W}\\colon \\mathscr{C}^{1}(H)\\to H$, called the Lie derivative with respect to $W$, defined by\n\t\\begin{equation}\n\t\t\\label{eq:Liederivative}\n\t\tL_W(v)=\\lim_{t\\to 0}\\frac{\\rho_{\\exp(tW)}(v)-v}{t}\\;, \\quad v\\in \\mathscr{C}^{1}(H).\n\t\\end{equation}\n\tObserve that, for a $\\mathscr{C}^{1}$-function $f$ on $M$, we have $L_W(f)=Wf$, regarding $\\mathscr{C}^{1}(M)$ as canonically embedded in $H$.\n\t\n\tThe center of the universal enveloping algebra $U(\\sl_2(\\R))$ is  generated, as an algebra, by any of its non-zero elements, called Casimir elements. For reasons to be clarified shortly, we choose the normalization of the Casimir element to be $\\square=-X^{2}+X+UV$, where $\\{X,U,V\\}$ is identified with a set of generators of the $\\R$-algebra $U(\\sl_2(\\R))$. By the universal property of the universal enveloping algebra, the Casimir element $\\square$ acts as a second-order linear differential operator $\\mathscr{C}^{2}(M)\\to \\mathscr{C}(M)$, and (in a compatible manner) as an unbounded operator on $H$, densely defined on the subspace $\\mathscr{C}^{2}(H)$. Slightly abusing notation, we shall again indicate with $\\square$ the Casimir operator arising in this fashion. It is a consequence of the Casimir element lying in the center of $U(\\sl_2(\\R))$ that the operator $\\square$ commutes with all unitary operators $\\rho_g,\\;g\\in \\SL_2(\\R)$ and all Lie derivatives $L_{W},\\;W\\in \\sl_2(\\R)$, wherever they are simultaneously defined.\n\t\n\tThe Casimir operator $\\square$ is an essentially self-adjoint operator on $H$; as $M$ is compact, its spectrum $\\text{Spec}(\\square)$ consists solely of eigenvalues and is a discrete subset of $\\R$. Our choice of the normalization of the Casimir element implies that, on the subspace $\\mathscr{C}^{2}(S)$ of continuously twice-differentiable functions defined on the surface $S=\\Gamma\\bsl \\Hyp$, the Casimir operator acts as the Laplace-Beltrami operator $\\Delta_{S}$ with respect to the hyperbolic Riemannian metric on $S$. As a consequence, the spectrum of $\\Delta_{S}$ is contained in the spectrum of $\\square$. As a matter of fact, it holds that $\\text{Spec}(\\Delta_S)=\\text{Spec}(\\square)\\cap \\R_{\\geq 0}$. \n\t\n\tA great deal of research has revolved around the properties of the spectrum of $\\Delta_S$; for the purposes of this article, we shall only need a weaker version (cf.~the proof of Lemma~\\ref{lem:finitespectralconstant}) of the celebrated Weyl's law concerning the asymptotics of the eigenvalues, which we shall now recall for completeness. \n\t\n\t\\begin{thm}[Weyl's law]\n\t\t\\label{thm:Weyllaw}\n\t\tLet $\\Gamma<\\SL_2(\\R)$ be as above,  $S=\\Gamma\\bsl \\Hyp$, and \n\t\t\\begin{equation*}\n\t\t\t\\lambda_0<\\lambda_1\\leq \\cdots \\leq \\lambda_n\\leq \\cdots \\to\\infty\n\t\t\\end{equation*} \n\t\tbe the eigenvalues, counted with multiplicity, of the Laplace-Beltrami operator $\\Delta_S$ associated to the hyperbolic Riemannian metric on $S$. Then\n\t\t\\begin{equation*}\n\t\t\t\\lim\\limits_{R\\to\\infty}\\frac{|\\{j\\in \\N:\\lambda_j\\leq R\\}|}{ R}=\\frac{\\emph{Area}(S)}{4\\pi}\\;,\n\t\t\\end{equation*}\n\t\twhere $\\emph{Area}(S)$ is the total mass of $S$ with respect to the hyperbolic area measure.\n\t\\end{thm}\n\tA proof of Theorem~\\ref{thm:Weyllaw} relying on Selberg's trace formula is given in~\\cite[Prop.~10]{Marklof}.\n\t\n\t\\medskip\n\tOn the other hand, the negative eigenvalues of $\\square$ are completely understood, being of the form $-m(m+2)\/4$, with $m$ ranging across the set $\\N^{\\ast}$ of strictly positive integers. \n\t\n\tIn general, any unitary representation of $\\SL_2(\\R)$ is unitarily equivalent to a direct integral of irreducible representations. Owing to compactness of $M$, the representation space $H$ of $\\rho$ actually decomposes as a Hilbert direct sum of (non-zero) irreducible $\\rho$-invariant subspaces:\n\t\\begin{equation*}\n\t\tH=\\bigoplus_{i \\in I}H_i\\;,\\quad \\rho(H_i)\\subset H_i,\\; \\rho|_{H_i} \\text{ irreducible},\\;I \\text{ countable index set}.\n\t\\end{equation*}\n\tAs the Casimir operator commutes with all the $\\rho_g$ for $g\\in \\SL_2(\\R)$, it restricts to an operator $\\mathscr{C}^2(H_i)\\to H_i$ for each $i \\in I$. By virtue of Schur's lemma for unbounded linear operators, there exists $\\mu \\in \\text{Spec}(\\square)$, depending on $i$, such that $\\square v=\\mu v$ for any $v\\in\\mathscr{C}^{2}(H_i)$. Regrouping all irreducible subspaces according to the corresponding Casimir eigenvalue, we obtain an orthogonal splitting into $\\rho$-invariant subspaces\n\t\\begin{equation}\n\t\t\\label{eq:deccasimireigenvalues}\n\t\tH=\\bigoplus\\limits_{\\mu \\in \\text{Spec}(\\square)}H_{\\mu}\\;,\\quad H_{\\mu}=\\{v\\in \\mathscr{C}^{2}(H):\\square v =\\mu v \\}.\n\t\\end{equation}\n\t\n\tLet $\\Theta\\in \\sl_2(\\R)$ be the element defined in~\\eqref{eq:rotationflow}, inducing the unbounded linear operator $L_{\\Theta}$ on $H$. As it is the case for any representation space of a unitary representation of $\\SL_2(\\R)$, each subspace $H_{\\mu}$ is the internal Hilbert direct sum \n\t\\begin{equation}\n\t\t\\label{eq:decthetaeigenvalues}\n\t\tH_{\\mu}=\\bigoplus_{n\\in \\Z}H_{\\mu,n}\\;,\\quad H_{\\mu,n}=\\biggl\\{v\\in \\mathscr{C}^{1}(H_{\\mu}):L_\\Theta v=\\frac{i}{2}nv \\biggr\\}.\n\t\\end{equation}  \n\t\n\t\\begin{rmk}\n\t\tObserve that, in the previous decomposition, it might occur (as it manifestly emerges in the equation~\\eqref{eq:differentSobolev} below) that $H_{\\mu,n}$ is the trivial subspace for some $\\mu\\in \\text{Spec}(\\square)$ and $n\\in \\Z$. For this reason, in the sequel, we let the index set of the direct sum in~\\eqref{eq:decthetaeigenvalues} be rather $I(\\mu)=\\{n\\in \\Z:H_{\\mu,n}\\neq \\{0\\} \\}\\subset \\Z$, bearing in mind this caveat.\n\t\\end{rmk}\n\t\n\t\\subsection{Sobolev spaces and the Sobolev Embedding Theorem}\n\t\\label{sec:Sobolev}\n\t\n\t\n\tThe classical theory of Sobolev spaces is thoroughly discussed in~\\cite{Adams}. In the context of arbitrary Riemannian manifolds, it is presented, for instance, in~\\cite{Aubin,Hebey}; the extension to fractional Sobolev orders is treated e.g.~in~\\cite{Strichartz-Sobolev,Triebel}. Here we shall confine ourselves to homogeneous spaces of $\\SL_2(\\R)$, placing emphasis on how this theory is brought to bear on the study of unitary representations of the special linear group. \n\t\n\tDefine the element\n\t\\begin{equation*}\n\t\tY=\n\t\t\\begin{pmatrix}\n\t\t\t0&-1\/2\\\\\n\t\t\t-1\/2&0\n\t\t\\end{pmatrix}\n\t\t\\in \\sl_2(\\R),\n\t\\end{equation*}\n\tso that $\\{X,\\Theta,Y \\}$ forms a basis of the real vector space $\\sl_2(\\R)$.\n\tDefine the second-order linear operator $\\Delta= -(X^{2}+Y^{2}+\\Theta^{2}) =\\square -2\\Theta^{2}$ on the subspace $\\mathscr{C}^{2}(H)$. As each Lie derivative $L_W$ ($W\\in \\sl_2(\\R)$) satisfies\\footnote{This follows at once from the definition of $L_W$ in~\\eqref{eq:Liederivative} and the fact that $\\rho_g$ is a unitary operator for any $g\\in \\SL_2(\\R)$.} $\\langle L_W u,v \\rangle =-\\langle u,L_Wv\\rangle$ for any $u,v \\in \\mathscr{C}^{1}(H)$, it is clear that $\\langle \\Delta u,v\\rangle =\\langle u, \\Delta v\\rangle$ for any $u,v\\in \\mathscr{C}^{2}(H)$, that is, $\\Delta$ is self-adjoint on its domain of definition. We shall refer to it as the Laplace operator on $H$. Akin to the Casimir operator, $\\Delta$ acts as the Laplace-Beltrami operator $\\Delta_S$ on the subspace $\\mathscr{C}^{2}(S)$. Indeed, any vector $u \\in \\mathscr{C}^{2}(S)$ is invariant under the action of $\\SO_2(\\R)$ by right translations, and hence lies in the kernel of $L_{\\Theta}$; as a consequence, \n\t\\begin{equation*}\n\t\t\\Delta u=(\\square -2\\Theta^{2})u=\\square u-2L_{\\Theta}^{2}u=\\square u=\\Delta_S u\\;. \n\t\\end{equation*}\n\t\n\tFor any $s\\in \\R_{>0}$, we define the Sobolev space of order $s$ on $H$, denoted by $W^{s}(H)$, as the maximal linear subspace of $H$ on which the unbounded linear operator $\\Delta^{s\/2}$ can be defined, and endow it with the inner product given by\n\t\\begin{equation}\n\t\t\\label{eq:Sobolevinnerproduct}\n\t\t\\langle u,v\\rangle_{W^{s}}=\\langle (I+\\Delta)^{s}u,v\\rangle\\;, \\quad u,v \\in W^{s}(H),\n\t\\end{equation}\n\twhere $I$ denotes the identity operator on $H$.\n\tThis assignment turns $W^{s}(H)$ into a Hilbert space, whose associated norm is denoted by $\\norm{\\cdot }_{W^{s}}$. Similarly, we define the Sobolev spaces $W^{s}(H_{\\mu})$ and $W^{s}(H_{\\mu,n})$ for any $\\mu \\in \\text{Spec}(\\Z)$ and $n\\in I(\\mu)$. It is a fact that the decompositions in~\\eqref{eq:deccasimireigenvalues} and~\\eqref{eq:decthetaeigenvalues} induce analogous decompositions on the level of Sobolev spaces, namely there are orthogonal\\footnote{Clearly, we intend that the closed subspaces $W^{s}(H_{\\mu,n})$ are orthogonal with respect to the $W^{s}$-inner product defined in~\\eqref{eq:Sobolevinnerproduct}. } splittings\n\t\\begin{equation}\n\t\t\\label{eq:Sobolevdecomposition}\n\t\tW^{s}(H)=\\bigoplus_{\\mu \\in \\text{Spec}(\\square)}W^{s}(H_{\\mu})=\\bigoplus_{\\mu \\in \\text{Spec}(\\square)}\\bigoplus_{n\\in I(\\mu)}\\;W^{s}(H_{\\mu,n})\\;.\n\t\\end{equation}\n\t\n\tThe argument in Section~\\ref{sec:arbitraryfn} crucially hinges upon the following elementary relationship between Sobolev norms of different order:\n\t\n\t\\begin{lem}\n\t\t\\label{lem:diffSobolevnorms}\n\t\tLet $s\\in \\R_{>0},\\;k\\in \\N$, and assume $u\\in W^{s+k}(H_{\\mu,n})$ for some $\\mu \\in \\emph{Spec}(\\square)$ and $n \\in I(\\mu)$. Then\n\t\t\\begin{equation}\n\t\t\t\\label{eq:differentSobolev}\n\t\t\t\\norm{u}_{W^{s+k}}^2=\\biggl(1+\\mu+\\frac{n^2}{2}\\biggr)^{k}\\norm{u}_{W^s}^2\\;.\n\t\t\\end{equation}\n\t\\end{lem} \n\t\\begin{proof}\n\t\tSuppose $k=1$. We may write, using self-adjointness of $\\Delta$ with respect to the $L^{2}$-inner product,\n\t\t\\begin{equation*}\n\t\t\t\\norm{u}_{W^{s+1}}^{2}=\\langle u,u\\rangle_{W^{s+1}}=\\langle (I+\\Delta)^{s+1}u,u\\rangle=\\langle (I+\\Delta)^{s}u,(I+\\Delta)u\\rangle=\\langle(1+\\Delta)^{s}u,u\\rangle +\\langle (1+\\Delta)^{s}u,\\Delta u \\rangle\\;.\n\t\t\\end{equation*}\n\t\tBy the assumption on $u$, it holds that $\\Delta u=(\\square-2\\Theta^{2})u=\\mu-2(\\frac{i}{2}n)^{2}u=(\\mu+\\frac{n^2}{2})u$. Therefore, we infer\n\t\t\\begin{equation*}\n\t\t\t\\norm{u}_{W^{s+1}}^{2}=\\norm{u}_{W^{s}}^2+\\biggl(\\mu+\\frac{n^{2}}{2}\\biggr)\\langle (1+\\Delta)^{s}u,u\\rangle = \\biggl(1+\\mu+\\frac{n^{2}}{2}\\biggr)\\norm{u}_{W^{s}}^{2}\\;,\n\t\t\\end{equation*}\n\t\tas desired.\n\t\t\n\t\tThe statement for an arbitrary $k\\in \\N$ is immediately achieved arguing by induction.\n\t\\end{proof}\n\t\n\tObserve that~\\eqref{eq:differentSobolev} readily implies the following: if $u \\in W^{s+k}(H)$ for some $s\\in \\R_{>0}$ and $k\\in \\N$,   and \n\t\\begin{equation*}\n\t\tu=\\sum_{\\mu\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}u_{\\mu,n}\n\t\\end{equation*}\n\tis its decomposition into joint eigenvectors for $\\square$ and $\\Theta$ provided by~\\eqref{eq:Sobolevdecomposition}, then the larger the integer $k$ is, the faster the decay of the Sobolev norms $\\norm{u_{\\mu,n}}_{W^{s}}$ as $|n|$ and $|\\mu|$ tend to infinity. This phenomenon\\footnote{The counterpart of this relationship in classical Fourier analysis is well-known; the regularity of a function is closely interwoven with the decay rate at infinity of its Fourier coefficients.} is going to be essential in our estimates over the course of the proof of Theorem~\\ref{thm:mainexpandingtranslates}.\n\t\n\t\\smallskip\n\tWe conclude this section recalling a version for compact three-manifolds of the celebrated Sobolev Embedding Theorem, which will be sufficient for our purposes.\n\t\n\t\n\t\n\t\\begin{thm}[Sobolev Embedding Theorem]\n\t\t\\label{thm:Sobolevembedding}\n\t\tFor any $r\\in \\N$ and $s\\in \\R_{>0}$ fulfilling the inequality $s-r>3\/2$, there is a continuous embedding of $W^{s}(M)$ into the Banach space $\\mathscr{C}^{r}(M)$: in particular, there exists a constant $C_{r,s}>0$ such that \n\t\t\\begin{equation*}\n\t\t\t\\norm{f}_{\\mathscr{C}^{r}}\\leq C_{r,s}\\norm{f}_{W^{s}}\n\t\t\\end{equation*}\n\t\tfor every $f\\in W^{s}(M)$.\n\t\\end{thm}\n\t\n\t\\section{Reduction to an ordinary differential equation}\n\t\\label{sec:reductiontoODE}\n\tThis section presents the gist of the approach we pursue in order to prove Theorem~\\ref{thm:case_Theta_eigenfn}, which concerns the asymptotic behaviour of circle-arc averages of joint eigenfunctions of the operators $\\square$ and $\\Theta$ (cf.~Section~\\ref{sec:unitaryrepresentations}); the partial differential equations (classically known in the literature as eigenvalue equations) expressing the eigenfunction condition are here shown to give rise to ordinary differential equations for the corresponding circle averages, when the latter are seen as functions of the time parameter.\n\t\n\tWe fix a function $f\\colon M\\to \\C$ of class $\\cC^{2}$ and a parameter $\\theta\\in (0,4\\pi]$. Recall from~\\eqref{eq:ktheta} the definition of the averages $k_{f,\\theta}(p,t)$, for $p \\in M$ and $t\\in \\R$. As we shall work with a fixed, arbitrary base point $p \\in M$, we shall abbreviate, for notational convenience, $k_{f,\\theta}(p,t)$ with $k_{\\theta}(t)$ in the computations that follow.\n\t\n\tOur goal is to show that the function $k_{\\theta}(t)$ satisfies a second-order linear ODE. In the upcoming computations, the following lemma will be of use. For any left-invariant vector field $W\\in \\sl_2(\\R)$, we indicate with $(\\phi_t^{W})_{t\\in \\R}$ be the one-parameter flow on $M$ defined by $\\phi^{W}_t(\\Gamma g)=\\Gamma g \\exp{tW}$ for any $t\\in \\R$ and $g\\in \\SL_2(\\R)$. For any pair $Y,W\\in \\sl_2(\\R)$ and any point $q\\in M$, the derivative of the smooth curve $s\\mapsto \\phi_{t}^{Y}\\circ \\phi^{W}_s(q)$ (seen as a function from $\\R$ to the tangent bundle of $M$), where $t\\in \\R$ is fixed, is denoted by $\\frac{\\text{d}}{\\text{d}s}\\;\\phi_t^{Y}\\circ \\phi_s^{W}(q)$.\n\tLastly, if $W\\in \\sl_2(\\R)$ and $q\\in M$, we denote by $W_q$ the value at $q$ of the infinitesimal generator of the smooth flow $(\\phi_t^{W})_{t\\in \\R}$ on $M$.\n\t\\begin{lem}\n\t\t\\label{lem:shiftedderivatives}\n\t\tFor every $Y,W\\in \\sl_2(\\R)\\setminus \\{ 0\\}$ and every $q\\in M$, it holds\n\t\t\\begin{equation*}\n\t\t\t\\frac{\\emph{d}}{\\emph{d}s}\\;\\phi_t^{Y}\\circ \\phi^{W}_s(q)=\\Ad_{\\exp{(-tY)}}(W)_{\\phi^{Y}_t\\circ \\phi_s^{W}(q)}\\;.\n\t\t\\end{equation*} \n\t\\end{lem}\n\t\\begin{proof}\n\t\tIt follows from elementary algebraic manipulations, see~\\cite[Lem.~4]{Rav-arcs}.\n\t\\end{proof}\n\t\n\tWe may now state the main result of this section.\n\t\n\t\\begin{prop}\n\t\t\\label{prop:ODE}\n\t\tLet $\\mu \\in \\emph{Spec}(\\square)$, $n\\in \\Z$ and $f\\in \\cC^{2}(M)$ be a function satisfying $\\square f=\\mu f$, $\\Theta f=\\frac{i}{2}n f$. For every $p \\in M$ and $\\theta\\in (0,4\\pi]$, there is a bounded continuous function $G_{\\theta,n}f(p,\\cdot)\\colon \\R_{>0}\\to \\C$ such that the function $k_{\\theta}(t)=\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\emph{d}s$ satisfies the linear ordinary differential equation\n\t\t\\begin{equation}\n\t\t\t\\label{eq:ode}\n\t\t\tk''_{\\theta}(t)+k'_{\\theta}(t)+\\mu k_{\\theta}(t)=e^{-t}G_{\\theta,n}(p,t)\n\t\t\\end{equation}\n\t\tfor any $t>0$.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tFix $f$, $p$ and $\\theta$ as in the assumptions. Since $f$ is of class $\\cC^{2}$ on $M$, differentiation under the integral sign gives\n\t\t\\begin{equation*}\n\t\t\tk'_{\\theta}(t)=\\frac{1}{\\theta} \\int_0^{\\theta}Xf\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s,\\quad k''_{\\theta}(t)=\\frac{1}{\\theta} \\int_0^{\\theta}X^{2}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\n\t\t\\end{equation*}\n\t\tfor any $t\\in \\R$, as the geodesic flow $(\\phi_t^{X})_{t\\in \\R}$ on $M$ is generated by the vector field $X$. Therefore, the assumption $\\square f=\\mu f$, i.e.~$-X^{2}f+Xf-UVf=\\mu f$, translates readily into\n\t\t\\begin{equation*}\n\t\t\tk''_{\\theta}(t)-k'_{\\theta}(t)+\\mu k_{\\theta}(t)=-\\frac{1}{\\theta}\\int_0^{\\theta}UVf\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s\\;.\n\t\t\\end{equation*}\n\t\tAs $V=U-2\\Theta$, we have \n\t\t\\begin{equation}\n\t\t\t\\label{ode}\n\t\t\tk''_{\\theta}(t)-k'_{\\theta}(t)+\\mu k_{\\theta}(t)=\t-\\frac{1}{\\theta}\\int_0^{\\theta}U^{2}f\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s+\\frac{in }{\\theta}\\int_0^{\\theta}Uf\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s\n\t\t\\end{equation}\n\t\tby the assumption $\\Theta f=\\frac{i}{2}nf$.\n\t\n\t\n\t\n\t\n\tNow, by Stokes' theorem\\footnote{Here, it really boils down to the fundamental theorem of calculus.}, we get that\n\t\\begin{equation*}\n\t\tf\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-f\\circ \\phi_t^{X}(p)=\\int_{0}^{\\theta}\\frac{\\text{d}}{\\text{d}s}(f\\circ \\phi_t^{X}\\circ r_{s}(p))\\;\\text{d}s=\\int_0^{\\theta}\\text{d}f_{\\phi_t^{X}\\circ r_{s}(p)}\\biggl(\\frac{\\text{d}}{\\text{d}s}(\\phi_t^{X}\\circ r_{s}(p))\\biggr)\\;\\text{d}s\\;,\n\t\\end{equation*}\n\tthe latter equality following from the chain rule for differentials. Recalling that the flow $(r_s)_{s\\in \\R}$ is generated by the vector field $\\Theta$, Lemma~\\ref{lem:shiftedderivatives} delivers\n\t\\begin{equation*}\n\t\t\\frac{\\text{d}}{\\text{d}s}\\phi_t^{X}\\circ r_s(q)=\\Ad_{\\exp(-tX)}(\\Theta)_{\\phi_{t}^{X}\\circ r_s(p)}\\;,\n\t\\end{equation*}\n\tso that\n\t\\begin{equation*} \n\t\t\\begin{split}\n\t\t\tf\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-f\\circ \\phi_t^{X}(p)&=\n\t\t\t\\int_0^{\\theta}\\text{d}f_{\\phi_t^{X}\\circ r_{s}(p)}\\bigl((\\Ad_{\\exp{(-tX)}}(\\Theta))_{\\phi_t^{X}\\circ r_s(p)}\\bigr)\\;\\text{d}s\\\\\n\t\t\t&=\\int_0^{\\theta}\\text{d}f_{\\phi_t^{X}\\circ r_{s}(p)}\\bigl(((-\\sinh{t})U+e^{t}\\Theta))_{\\phi_t^{X}\\circ r_s(p)}\\bigr)\\;\\text{d}s\\\\\n\t\t\t&=-\\sinh{t}\\int_0^{\\theta}Uf\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s+\\frac{i}{2}n\\theta e^{t}k_{\\theta}(t)\\;,\n\t\t\\end{split}\n\t\\end{equation*}\n\tthe second equality being obtained by straightforward matrix multiplications.\n\t\n\tFrom now we choose $t$ strictly positive. We may thus write\n\t\\begin{equation}\n\t\t\\label{u}\n\t\t\\int_0^{\\theta}Uf\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s=\\frac{1}{\\sinh{t}}\\biggl(\\frac{i}{2}n\\theta e^{t}k_{\\theta}(t)-A_{\\theta}(t)\\biggr)\n\t\\end{equation}\n\twhere\n\t\\begin{equation*} A_{\\theta}(t)=f\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-f\\circ \\phi_t^{X}(p)\\;,\\quad t>0.\n\t\\end{equation*}\n\tArguing as before, we also deduce\n\t\\begin{equation}\n\t\t\\label{dedusquare}\n\t\t\\begin{split}\n\t\t\tUf\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-Uf\\circ \\phi_t^{X}(p)&=\\int_{0}^{\\theta}\\frac{\\text{d}}{\\text{d}s}(Uf\\circ \\phi_t^{X}\\circ r_{s}(p))\\;\\text{d}s\\\\%=\\int_{0}^{\\theta}\\text{d}Uf_{\\phi_t^{X}\\circ r_s(p)}\\bigl((\\Ad_{\\exp{(-tX)}}(\\Theta))_{\\phi_t^{X}\\circ r_s(p)}\\bigr)\\;\\text{d}s\\\\\n\t\t\t&=\\frac{1}{2}e^{-t}\\int_0^{\\theta}U^{2}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s-\\frac{1}{2}e^{t}\\int_0^{\\theta}VUf\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\\;.\n\t\t\\end{split}\n\t\\end{equation}\n\tFrom $UV-VU=2X$ we get $VU=UV-2X=U(U-2\\Theta)-2X=U^{2}-2U\\Theta-2X$, so that \n\t\\begin{equation}\n\t\t\\label{vu}\n\t\t\\int_0^{\\theta} VUf\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s=\\int_0^{\\theta} U^2f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s-in\\int_0^{\\theta}Uf\\circ \\phi_t^{X}\\circ r_s(p)\\text{d}s-2\\theta k'_{\\theta}(t)\\;.\n\t\\end{equation}\n\tCombining~\\eqref{u},~\\eqref{dedusquare} and~\\eqref{vu} yields\n\t\\begin{equation}\n\t\t\\label{usquare}\n\t\t\\begin{split}\n\t\t\t\\int_0^{\\theta}U^{2}f\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s&=\\frac{1}{\\sinh{t}}\\biggl(\\frac{i}{2}ne^{t}\\int_0^{\\theta}Uf\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s+\\theta e^{t}k'_{\\theta}(t) -B_{\\theta}(t)\\biggr)\\\\\n\t\t\t&=\\frac{1}{\\sinh{t}}\\biggl(\\frac{ine^{t}}{4\\sinh{t}}(in\\theta e^{t} k_{\\theta}(t)-A_{\\theta}(t)) +\\theta e^{t}k'_{\\theta}(t) -B_{\\theta}(t)\\biggr)\\;,\n\t\t\\end{split}\n\t\\end{equation}\n\twhere\n\t\\begin{equation*} B_{\\theta}(t)=Uf\\circ\\phi_{t}^{X}\\circ r_{\\theta}(p)-Uf\\circ \\phi_t^{X}(p)\\;,\\quad t>0.\n\t\\end{equation*}\n\tFrom~\\eqref{ode},~\\eqref{u} and~\\eqref{usquare} we infer that \n\t\\begin{equation*}\n\t\tk_{\\theta}''(t)-k'_{\\theta}(t)+\\mu k_{\\theta}(t)=\\frac{n^2 e^{2t} k_{\\theta}(t)}{4\\sinh^2{t}}+\\frac{ine^{t}A_{\\theta}(t)}{4\\theta\\sinh^2{t}} - \\frac{e^{t}k'_{\\theta}(t)}{\\sinh{t}} -\\frac{n^2 e^{t}k_{\\theta}(t)}{2\\sinh{t}}+\\frac{-inA_{\\theta}(t)+B_{\\theta}(t)}{\\theta \\sinh{t}}\n\t\\end{equation*}\n\tFinally, adding $2k_{\\theta}'(t)$ on both sides gives\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\tk_{\\theta}''(t)+k'_{\\theta}(t)+\\mu k_{\\theta}(t)&=\\biggl(\\frac{n^{2}}{(1-e^{-2t})^{2}}-\\frac{n^{2}}{1-e^{-2t}}\\biggr)k_{\\theta}(t)+\\biggl(2-\\frac{2}{1-e^{-2t}}\\biggr)k'_{\\theta}(t)\\\\\n\t\t\t&+\\biggl(\\frac{in}{2\\theta \\sinh{t}(1-e^{-2t})}-\\frac{in}{\\theta\\sinh{t}}\\biggr)A_{\\theta}(t)+\\frac{B_{\\theta}(t)}{\\theta \\sinh{t}}\\;,\n\t\t\\end{split}\n\t\\end{equation*}\n\tso that $k_{\\theta}''(t)+k'_{\\theta}(t)+\\mu k_{\\theta}(t)=e^{-t}G_{\\theta,n}f(p,t)$ for \n\t\\begin{equation}\n\t\t\\label{eq:constantterm}\n\t\tG_{\\theta,n}f(p,t)=\\frac{n^{2}e^{-t}}{(1-e^{-2t})^{2}}k_{\\theta}(t)-\\frac{2e^{-t}}{1-e^{-2t}}k'_{\\theta}(t)+\\frac{2ine^{-2t}}{\\theta(1-e^{-2t})^{2}}A_{\\theta}(t)+\\frac{2}{\\theta (1-e^{-2t})}B_{\\theta}(t)\\;.\n\t\\end{equation}\n\tThe function $f$ being of class $\\cC^{2}$, it is clear that the functions $k_{\\theta},k'_{\\theta},A_{\\theta}$ and $B_{\\theta}$ are continuous, hence so is the function $t\\mapsto G_{\\theta,n}f(p,t)$.  Furthermore, the trivial upper bounds\n\t\\begin{equation*}\n\t |k_{\\theta}(t)|\\leq \\norm{f}_{\\infty},\\;|k'_{\\theta}(t)|\\leq \\norm{Xf}_{\\infty},\\;|A_{\\theta}(t)|\\leq 2\\norm{f}_{\\infty},\\; |B_{\\theta}(t)|\\leq 2\\norm{Uf}_{\\infty}\n\t \\end{equation*}\n\t  imply that it is uniformly bounded on $\\R_{>0}$.\n\\end{proof}\nFor later purposes, we estimate explicitly the uniform norm of $G_{\\theta,n}f$. Using the bounds $e^{-t}\\leq 1$ and $1-e^{-2t}\\geq 1-e^{-1}$, valid for all $t\\geq 1\/2$, together with the fact that the three quantities $\\norm{f}_{\\infty},\\norm{Xf}_{\\infty},\\norm{Uf}_{\\infty}$ are bounded from above by $\\norm{f}_{\\mathscr{C}^{1}}$ (cf.~Section~\\ref{sec:hyperbolic}), we obtain that \n\\begin{equation*}\n\t\\sup_{t\\geq 1\/2}|G_{\\theta,n}f(p,t)|\\leq C_{\\theta,n}\\norm{f}_{\\mathscr{C}^{1}}\n\\end{equation*}\nwith\n\\begin{equation*}\n\tC_{\\theta,n}=\\biggl(\\frac{e}{e-1}\\biggr)^{2}\\frac{n(\\theta n+2)}{\\theta}+\\frac{e}{e-1}\\frac{2\\theta+2}{\\theta}\\;.\n\\end{equation*}\nSetting\n\\begin{equation*} \\kappa_0=\\frac{2e^{2}(1+4\\pi)}{(e-1)^{2}}\\;,\n\\end{equation*}\nwe may estimate\n\\begin{equation*}\n\tC_{\\theta,n}\\leq \\biggl(\\frac{e}{e-1}\\biggr)^{2}\\;\\frac{2(\\theta+1)}{\\theta}(n^{2}+1)\\leq \\frac{\\kappa_0}{\\theta}(n^{2}+1)\\;,\n\\end{equation*}\nand hence conclude that \n\\begin{equation}\n\t\\label{eq:Gtheta}\n\t\\sup_{t\\geq 1\/2}|G_{\\theta,n}f(p,t)|\\leq\\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\cC^{1}}\n\\end{equation}\nfor every choice of $p \\in M$, $\\theta\\in (0,4\\pi]$, $f\\in \\cC^{2}(M)$ and $n\\in \\Z$.\n\n\n\\section{Asymptotics for joint eigenfunctions}\n\\label{sec:jointeigenfn}\n\nThe purpose of this section is to prove Theorem~\\ref{thm:case_Theta_eigenfn} by explictly solving the ODE established in Proposition~\\ref{prop:ODE}. For definiteness, whe choose to impose initial conditions at time $t=1$ for the ensuing Cauchy problem. \n\nWe thus start with the following:\n\n\\begin{lem}\n\t\\label{lem:solution}\n\tLet $\\mu$ be an eigenvalue of the Casimir operator. If $G\\colon \\R_{>0}\\to \\C$ is a continuous function, then for any complex numbers $y_1$ and $y_1'$ the solution to the Cauchy problem\n\t\\begin{equation}\n\t\t\\label{eq:Cauchy}\n\t\t\\begin{cases}\n\t\t\ty''(t)+y'(t)+\\mu y(t)=e^{-t}G(t) & \\\\\n\t\t\ty(1)=y_1& \\\\\n\t\t\ty'(1)=y'_1\n\t\t\\end{cases}\n\t\\end{equation}\n\tis given by\n\t\\begin{equation}\n\t\t\\label{eq:solutionnotquarter}\n\t\t\\begin{split}\n\t\t\ty(t)=&e^{-\\frac{1-\\nu}{2}t}\\biggl(\\frac{(1+\\nu)y_1+2y'_1}{2\\nu e^{-\\frac{1-\\nu}{2}}} +\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1+\\nu}{2}\\xi}G(\\xi)\\;\\emph{d}\\xi \\biggr)\\\\\n\t\t\t&-e^{-\\frac{1+\\nu}{2}t}\\biggl(\\frac{(1-\\nu)y_1+2y'_1}{2\\nu e^{-\\frac{1+\\nu}{2}}} +\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1-\\nu}{2}\\xi}G(\\xi)\\;\\emph{d}\\xi \\biggr)\n\t\t\\end{split}\n\t\\end{equation}\n\tif $\\mu\\neq 1\/4$, and by\n\t\\begin{equation}\n\t\t\\label{eq:solutionquarter}\n\t\t\\begin{split}\n\t\t\ty(t)=&e^{-\\frac{t}{2}}\\biggl(\\frac{\\sqrt{e}(y_1-2y'_1)}{2}-\\int_1^{t}\\xi e^{-\\frac{\\xi}{2}}G(\\xi)\\;\\emph{d}\\xi\\biggr)\\\\\n\t\t\t& +te^{-\\frac{t}{2}}\\biggl(\\frac{\\sqrt{e}(y_1+2y'_1)}{2}+\\int_1^{t} e^{-\\frac{\\xi}{2}}G(\\xi)\\;\\emph{d}\\xi\\biggr)\n\t\t\\end{split}\n\t\\end{equation}\n\tif $\\mu=1\/4$.\n\\end{lem}\n\\begin{proof}\n\tLet $\\nu$ be the unique complex number in $\\R_{\\geq 0}\\cup i \\R_{>0}$ such that $1-\\nu^{2}=4\\mu$.\n\tThe characteristic polynomial of the homogeneous equation $y''(t)+y'(t)+\\mu y(t)=0$ is $P(Z)=Z^{2}+Z+\\mu$, having two distinct roots $-\\frac{1-\\nu}{2},-\\frac{1+\\nu}{2}$ if $\\mu\\neq 1\/4$, and a double root $-\\frac{1}{2}$ if $\\mu=1\/4$. We examine the case $\\mu\\neq 1\/4$; the case $\\mu=1\/4$ requires only minor modifications. A particular solution of the inhomogeneous equation is given by\n\t\\begin{equation*}\n\t\te^{-\\frac{1-\\nu}{2}t} +\\int_1^{t}\\frac{1}{\\nu}e^{-\\frac{1+\\nu}{2}\\xi}G(\\xi)\\;\\text{d}\\xi\n\t\t-e^{-\\frac{1+\\nu}{2}t}\\int_1^{t}\\frac{1}{\\nu}e^{-\\frac{1-\\nu}{2}\\xi}G(\\xi)\\;\\text{d}\\xi\n\t\\end{equation*}\n\tas direct computations allow to verify. Hence, the general solution of\n\t\\begin{equation*} y''(t)+y'(t)+\\mu y(t)=e^{-t}G(t)\n\t\\end{equation*}\n\tis given by\n\t\\begin{equation*}\n\t\ty(t)=e^{-\\frac{1-\\nu}{2}t}\\biggl(c_1 +\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1+\\nu}{2}\\xi}G(\\xi)\\;\\text{d}\\xi \\biggr)\n\t\t+e^{-\\frac{1+\\nu}{2}t}\\biggl(c_2 -\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1-\\nu}{2}\\xi}G(\\xi)\\;\\text{d}\\xi \\biggr)\\;, \\quad c_1,c_2\\in \\C.\n\t\\end{equation*}\n\tImposing the conditions $y(1)=y_1,y'(1)=y'_1$ enables to determine the coefficients\n\t\\begin{equation*}\n\t\tc_1=\\frac{(1+\\nu)y_1+2y'_1}{2\\nu e^{-\\frac{1-\\nu}{2}}}\\;,\\quad c_2=-\\frac{(1-\\nu)y_1+2y'_1}{2\\nu e^{-\\frac{1+\\nu}{2}}}\\;.\n\t\\end{equation*} \n\\end{proof}\n\nWe may now apply Proposition~\\ref{prop:ODE} in conjunction with Lemma~\\ref{lem:solution} to determine the explicit analytic expression of the function $k_{f,\\theta}(p,t)$ (cf.~\\eqref{eq:ktheta}) in terms of the coefficient $G_{\\theta,n}f(p,t)$ (cf.~\\eqref{eq:constantterm}).\nBefore proceeding with this,  it will be convenient to set some useful notation first.\n\nLet $\\mu\\in \\text{Spec}(\\square)$, $f\\in \\cC^{2}(M)$ and $\\theta\\in (0,4\\pi]$. Define the functions $a_{\\theta,\\mu}^{+},a_{\\theta,\\mu}^{-}f\\colon M\\to \\C$ by\n\\begin{equation}\n\t\\label{eq:anquarter}\n\ta^{\\pm}_{\\theta,\\mu}f(p)= \\mp\\frac{(1\\mp\\nu)\\theta^{-1}\\int_0^{\\theta}f\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s+2\\theta^{-1}\\int_0^{\\theta}Xf\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s}{2\\nu e^{-\\frac{1\\pm\\nu}{2}}}\n\\end{equation}\nif $\\mu\\neq 1\/4$ and\n\\begin{equation}\n\t\\label{eq:aquarter}\n\ta^{\\pm}_{\\theta,1\/4}f(p)= \\frac{\\sqrt{e}\\bigl(\\theta^{-1}\\int_0^{\\theta}f\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s\\mp 2\\theta^{-1}\\int_0^{\\theta}Xf\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s\\bigr)}{2}\n\\end{equation}\nif $\\mu=1\/4$. When $\\mu\\neq 1\/4$, it holds \n\\begin{equation}\n\t\\label{eq:aestnquarter}\n\t\\norm{a^{\\pm}_{\\theta,\\mu}f}_{\\infty}\\leq \\frac{(1+|\\nu|)\\theta^{-1}\\int_0^{\\theta}\\norm{f}_{\\infty}\\text{d}s+2\\theta^{-1}\\int_0^{\\theta}\\norm{Xf}_{\\infty}\\text{d}s}{2e^{-1}|\\nu|}\\leq  \\frac{e(3+|\\nu|)}{2|\\nu|}\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\\end{equation}\nsince $\\norm{f}_{\\infty}\\leq \\norm{f}_{\\mathscr{C}^{1}}$ and $\\norm{Xf}_{\\infty}\\leq \\norm{f}_{\\mathscr{C}^{1}}$.\nIf $\\mu=1\/4$, similar estimates lead readily to\n\\begin{equation}\n\t\\label{eq:aestquarter}\n\t\\norm{a^{\\pm}_{\\theta,1\/4}f}_{\\infty}\\leq \\frac{3\\sqrt{e}}{2}\\norm{f}_{\\mathscr{C}^{1}}\\;.\n\\end{equation}\n\nWe are now in a position to start the proof of Theorem~\\ref{thm:case_Theta_eigenfn}, which will occupy the remainder of this section. We fix $\\theta\\in (0,4\\pi]$ and a function $f\\in \\cC^{2}(M)$ satisfying $\\Theta f=\\mu f$ and $\\Theta f=\\frac{i}{2}nf$ for some $\\mu \\in \\text{Spec}(\\square)$ and $n\\in \\Z$. For any $p \\in M$, the function $k_{f,\\theta}(p,\\cdot)\\colon \\R_{>0}\\to \\C$ we are interested in satisfies~\\eqref{eq:Cauchy} with initial conditions\n\\begin{equation*}\n\ty_1=\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_1^{X}\\circ r_s(p)\\;\\text{d}s\\;,\\quad y'_1=\\frac{1}{\\theta}\\int_0^{\\theta}Xf\\circ \\phi_1^{X}\\circ r_s(p)\\;\\text{d}s\\;.\n\\end{equation*} \n\nWe distinguish five cases as in the statement of Theorem~\\ref{thm:case_Theta_eigenfn}, that is, according to the value of the Casimir eigenvalue $\\mu$. Recall that $\\nu$ is the unique complex number in $\\R_{\\geq 0}\\cup i \\R_{>0}$ verifying $1-\\nu^{2}=4\\mu$.\n\n\\subsection{The case $\\mu>1\/4$}\n\\label{sec:caseabovequarter}\nSuppose $\\mu>1\/4$, so that $\\nu=i\\Im{\\nu}\\in i\\R_{>0}$. As follows from~\\eqref{eq:solutionnotquarter}, the solution to~\\eqref{eq:ode} with the prescribed initial conditions is given by\n\\begin{equation*}\n\t\\begin{split}\n\t\tk_{f,\\theta}(p,t)&=e^{-\\frac{t}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\biggl(a^{+}_{\\theta,\\mu}f(p)+a^{-}_{\\theta,\\mu}f(p)-\\frac{2}{\\Im{\\nu}}\\int_1^{t}e^{-\\frac{\\xi}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&+e^{-\\frac{t}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\biggl(a^{-}_{\\theta,\\mu}f(p)-a^{+}_{\\theta,\\mu}f(p)-\\frac{2i}{\\Im{\\nu}}\\int_1^{t}e^{-\\frac{\\xi}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\t\\end{split}\n\\end{equation*}\n   The functions\n   \\begin{equation*}\n    e^{-\\xi\/2}\\cos{(\\frac{\\Im{\\nu}}{2}\\xi)}G_{\\theta,n}f(p,\\xi),\\quad e^{-\\xi\/2}\\sin{(\\frac{\\Im{\\nu}}{2}\\xi)}G_{\\theta,n}f(p,\\xi) \n   \\end{equation*} \n    are integrable over the closed half-line $[1,+\\infty)$, as $G_{\\theta,n}f(p,\\cdot)$ is bounded thereon; we may therefore define functions $D^{+}_{\\theta,\\mu,n}f,D^{-}_{\\theta,\\mu,n}f\\colon M\\to \\C$ by setting\n\\begin{equation}\n\t\\label{eq:Dplusabovequarter}\n\tD^{+}_{\\theta,\\mu,n}f(p)=a^{+}_{\\theta,\\mu}f(p)+a^{-}_{\\theta,\\mu}f(p)-\\frac{2}{\\Im{\\nu}}\\int_1^{\\infty}e^{-\\frac{\\xi}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\\end{equation}\nand \n\\begin{equation}\n\t\\label{eq:Dminusabovequarter}\n\tD^{-}_{\\theta,\\mu,n}f(p)=a^{-}_{\\theta,\\mu}f(p)-a^{+}_{\\theta,\\mu}f(p)-\\frac{2i}{\\Im{\\nu}}\\int_1^{\\infty}e^{-\\frac{\\xi}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\\end{equation}\nfor every $p \\in M$. Then~\\eqref{eq:estabovequarter} is valid with \n\\begin{equation}\n\t\\label{eq:remainderabovequarter}\n\t\\begin{split}\n\t\t\\mathcal{R}_{\\theta,\\mu,n}f(p,t)&=e^{-\\frac{t}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\int_t^{\\infty}\\frac{2}{\\Im{\\nu}}e^{-\\frac{\\xi}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\\\\n\t\t&+e^{-\\frac{t}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\int_t^{\\infty}\\frac{2i}{\\Im{\\nu}}e^{-\\frac{\\xi}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\;,\\quad t\\geq 1, \\;p \\in M.\n\t\\end{split}\n\\end{equation}\nLet us now estimate the uniform norms of $D^{\\pm}_{\\theta,\\mu,n}f$ and of $\\mathcal{R}_{\\theta,\\mu,n}f(\\cdot,t)$ for any $t\\geq 1$. From the explicit expressions in~\\eqref{eq:Dplusabovequarter} and~\\eqref{eq:Dminusabovequarter}, it follows at once that\n\\begin{equation}\n\t\\label{eq:kappamu}\n\t\\begin{split}\n\t\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f}_{\\infty}&\\leq \\norm{a^{+}_{\\theta,\\mu}f}_{\\infty}+\\norm{a^{-}_{\\theta,\\mu}}_{\\infty}+\\frac{2}{\\Im{\\nu}}\\int_1^{\\infty}e^{-\\frac{\\xi}{2}}\\sup_{p\\in M,\\;\\xi\\geq 1}|G_{\\theta,n}f(p,\\xi)|\\;\\text{d}\\xi\\\\\n\t\t&\\leq \\frac{1}{\\Im{\\nu}}\\biggl(e(3+\\Im{\\nu})+\\frac{4\\kappa_0(n^{2}+1)}{\\theta\\sqrt{e}}\\biggr)\\norm{f}_{\\mathscr{C}^{1}}\\\\\n\t\t&\\leq\\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\n\t\\end{split}\n\\end{equation}\napplying~\\eqref{eq:Gtheta} and~\\eqref{eq:aestnquarter} in the second inequality, with $\\kappa(\\mu)=\\frac{1}{\\Im{\\nu}}\\bigl(4e\\pi(3+\\Im{\\nu})+\\frac{4\\kappa_0}{\\sqrt{e}}\\bigr)$.\n\nThe remainder term defined in~\\eqref{eq:remainderabovequarter} is bounded from above by\n\\begin{equation*}\n\t|\\mathcal{R}_{\\theta,\\mu,n}f(p,t)|\\leq 2e^{-\\frac{t}{2}}\\sup_{p \\in M,\\;\\xi\\geq 1}|G_{\\theta,n}f(p,\\xi)|\\int_t^{\\infty}\\frac{2}{\\Im{\\nu}}e^{-\\frac{\\xi}{2}}\\;\\text{d}\\xi\\leq\\frac{8\\kappa_0 (n^{2}+1)}{\\theta\\;\\Im{\\nu}}\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\n\\end{equation*}\nfor every $p\\in M$ and $t\\geq 1$, again relying on the upper bound in~\\eqref{eq:Gtheta}. \n\n\\smallskip\nUp to the regularity claims on the coefficients $D^{\\pm}_{\\theta,\\mu,n}f$, which are the subject of Section~\\ref{sec:regularity}, the proof of Theorem~\\ref{thm:case_Theta_eigenfn}(1) is complete.  \n\n\\subsection{The case $\\mu=1\/4$}\n\nSuppose $\\mu=1\/4$, whence $\\nu=0$. This time the solution to~\\eqref{eq:ode} with the given initial conditions has the expression (cf.~\\eqref{eq:solutionquarter}) \n\\begin{equation*}\n\n\tk_{f,\\theta}(p,t)=e^{-\\frac{t}{2}}\\biggl(a^{+}_{\\theta,1\/4}f(p)-\\int_1^{t}\\xi e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\n\t+te^{-\\frac{t}{2}}\\biggl(a^{-}_{\\theta,1\/4}f(p)+\\int_1^{t} e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\\end{equation*}\nFollowing the steps carried out in Section~\\ref{sec:caseabovequarter} almost verbatim, define functions $D^{+}_{\\theta,1\/4,n}, D^{-}_{\\theta,1\/4,n}\\colon M\\to \\C$ via\n\\begin{equation}\n\t\\label{eq:Dquarter}\n\tD^{+}_{\\theta,1\/4,n}f(p)=a^{+}_{\\theta,1\/4}f(p)-\\int_1^{\\infty}\\xi e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi,\\quad D^{-}_{\\theta,1\/4,n}f(p)=a^{-}_{\\theta.1\/4}f(p)+\\int_1^{\\infty} e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\\end{equation}\nfor every $p \\in M$. Then~\\eqref{eq:estquarter} holds with \n\\begin{equation}\n\t\\label{eq:remainderquarter}\n\t\\mathcal{R}_{\\theta.1\/4,n}f(p,t)=e^{-\\frac{t}{2}}\\int_t^{\\infty}\\xi e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi -te^{-\\frac{t}{2}}\\int_t^{\\infty}e^{-\\frac{\\xi}{2}}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\;, \\quad t\\geq 1,\\;p\\in M.\n\\end{equation}\nFrom~\\eqref{eq:Dquarter} we estimate, by virtue of~\\eqref{eq:Gtheta} and~\\eqref{eq:aestquarter}, \n\\begin{equation*}\n\t\\begin{split}\n\t\t\\norm{D^{\\pm}_{\\theta,1\/4,n}f}_{\\infty}&\\leq \\frac{3\\sqrt{e}}{2}\\norm{f}_{\\mathscr{C}^{1}}+\\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\int_1^{\\infty}\\xi e^{-\\frac{\\xi}{2}}\\;\\text{d}\\xi\\\\\n\t\t&=\\biggl(\\frac{3\\sqrt{e}}{2}+\\frac{6\\kappa_0}{\\theta\\sqrt{e}}(n^{2}+1)\\biggr)\\norm{f}_{\\mathscr{C}^{1}}\\leq \\frac{\\kappa(1\/4)}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\t\\end{split}\n\\end{equation*}\nwhere we may choose $\\kappa(1\/4)=36\\pi \\sqrt{e}\\kappa_0$. Moreover, we deduce from~\\eqref{eq:remainderquarter} that, for every $p \\in M$ and $t\\geq 1$,\n\\begin{equation*}\n\t\\begin{split}\n\t\t|\\mathcal{R}_{\\theta,1\/4,n}f(p,t)|&\\leq \\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\biggl(e^{-\\frac{t}{2}}\\int_t^{\\infty}\\xi e^{-\\frac{\\xi}{2}}\\;\\text{d}\\xi+te^{-\\frac{t}{2}}\\int_t^{\\infty}e^{-\\frac{\\xi}{2}}\\;\\text{d}\\xi\\biggr) \\\\ \n\t\t&=\\frac{4\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}(t+1)e^{-t}\\;.\n\t\\end{split}\n\\end{equation*}\nIn conjunction with Section~\\ref{sec:regularity}, this concludes the proof of Theorem~\\ref{thm:case_Theta_eigenfn}(2).\n\n\\subsection{The case $0<\\mu<1\/4$} When $0<\\mu<1\/4$, we have $\\nu\\in (0,1)$. The solution to~\\eqref{eq:ode} is given, as in~\\eqref{eq:solutionnotquarter}, by\n\\begin{equation*}\n\t\\begin{split}\n\t\tk_{f,\\theta}(p,t)=&e^{-\\frac{1+\\nu}{2}t}\\biggl(a^{+}_{\\theta,\\mu}f(p)-\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1-\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&+e^{-\\frac{1-\\nu}{2}t}\\biggl(a^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{1+\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\t\\end{split}\n\\end{equation*}\nSetting\n\\begin{equation}\n\t\\label{eq:Dbelowquarter}\n\t\\begin{split}\n\t\t&D^{+}_{\\theta,\\mu,n}f(p)=a^{+}_{\\theta,\\mu}f(p)-\\frac{1}{\\nu}\\int_1^{\\infty}e^{-\\frac{1-\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\;, \\\\ &D^{-}_{\\theta,\\mu,n}f(p)=a^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{\\infty}e^{-\\frac{1-\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\t\\end{split}\n\\end{equation}\nand \n\\begin{equation*}\n\t\\mathcal{R}_{\\theta,\\mu,n}f(p,t)=\\frac{1}{\\nu}\\biggl(e^{-\\frac{1+\\nu}{2}t}\\int_t^{\\infty}e^{-\\frac{1-\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi-e^{-\\frac{1-\\nu}{2}t}\\int_t^{\\infty}e^{-\\frac{1+\\nu}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\n\\end{equation*}\nfor every $p \\in M$ and $t\\geq 1$, it is clear that~\\eqref{eq:estpositivebelowquarter} holds. As far as estimates on the supremum norm are concerned, we have\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f}_{\\infty}&\\leq \\frac{e(3+\\nu)}{2\\nu}\\norm{f}_{\\mathscr{C}^{1}}+\\frac{\\kappa_0}{\\theta\\nu}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\int_1^{\\infty}e^{-\\frac{1\\mp \\nu}{2}\\xi}\\;\\text{d}\\xi\\\\\n\t\t&=\\frac{1}{\\nu}\\biggl(\\frac{e(3+\\nu)}{2}+\\frac{\\kappa_0}{\\theta}\\frac{2}{1\\mp \\nu}e^{-\\frac{1\\mp \\nu}{2}}(n^{2}+1)\\biggr)\\norm{f}_{\\mathscr{C}^{1}}\\\\\n\t\t&\\leq \\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\t\\end{split}\n\\end{equation*}\nfor $\\kappa(\\mu)=\\frac{2\\kappa_0e^{-\\frac{1-\\nu}{2}}}{\\nu(1-\\nu)}+\\frac{2e\\pi(3+\\nu)}{\\nu}$, and\n\\begin{equation*}\n\t\\begin{split}\n\t\t|\\mathcal{R}_{\\theta,\\mu,n}f(p,t)|&\\leq \\frac{\\kappa_0}{\\theta\\nu}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\biggl(e^{-\\frac{1+\\nu}{2}t}\\int_t^{\\infty}e^{-\\frac{1-\\nu}{2}\\xi}\\;\\text{d}\\xi+e^{-\\frac{1-\\nu}{2}t}\\int_t^{\\infty}e^{-\\frac{1+\\nu}{2}\\xi}\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&= \\frac{4\\kappa_0}{\\theta \\nu(1-\\nu)(1+\\nu)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;,\n\t\\end{split}\n\\end{equation*}\nwhich achieves the proof of Theorem~\\ref{thm:case_Theta_eigenfn}(3) except for the regularity of the coefficients which is addressed separately in Section~\\ref{sec:regularity}.\n\n\\subsection{The case $\\mu= 0$} As $\\nu=1$ when $\\mu=0$,  equation~\\eqref{eq:solutionnotquarter} delivers the following expression for the solution to~\\eqref{eq:ode}:\n\\begin{equation}\n\t\\label{eq:kzero}\n\t\\begin{split}\n\t\tk_{f,\\theta}(p,t)&=a^{-}_{\\theta,0}f(p)+\\int_1^{t}e^{-\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi+e^{-t}\\biggl(a^{+}_{\\theta,0}f(p)-\\int_1^{t}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&= a^{-}_{\\theta,0}f(p)+\\int_1^{\\infty}e^{-\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi+\\biggl(-\\int_t^{\\infty}e^{-\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\\\\n\t\t& +e^{-t}a^{+}_{\\theta,0}f(p)-e^{-t}\\int_1^{t}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\nObserve that the term between parentheses in the last expression is infinitesimal as $t$ tends to infinity, so that $k_{f,\\theta}(p,t)$ has a well-defined limit, as $t$ tends to infinity, for every $p \\in M$.\nWe claim that\\footnote{The claim amounts to the qualitative equidistribution statement that circle-arc averages of $f$ converge to its spatial average with respect to the uniform measure. This has been shown by Margulis (for complete circles) in~\\cite{Margulis} via a thickening argument resting on the mixing properties of the geodesic flow. We prefer not to invoke Margulis' result here, and instead prove directly this special case of equidistribution using spectral considerations coupled with mixing.  }    \n\\begin{equation}\n\t\\label{eq:constanttermzero}\n\t\\lim_{t\\to\\infty}k_{f,\\theta}(p,t)=\\int_{M}f\\;\\text{d}\\vol\n\\end{equation}\nfor any $p \\in M$. For a start, we show the equality in~\\eqref{eq:constanttermzero} holds on average with respect to the measure $\\vol$. Indeed, Fubini's theorem gives, for any $t>0$,\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\int_{M}k_{f,\\theta}(p,t)\\;\\text{d}\\vol(p)&=\\frac{1}{\\theta}\\int_{M}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\\;\\text{d}\\vol(p)=\\frac{1}{\\theta}\\int_{0}^{\\theta}\\int_{M}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}\\vol(p)\\;\\text{d}s\\\\\n\t\t&=\\frac{1}{\\theta}\\int_0^{\\theta}\\int_{M}f\\;\\text{d}\\vol\\;\\text{d}s=\\int_{M}f\\;\\text{d}\\vol\\;,\n\t\\end{split}\n\\end{equation*}\nthe second-to-last equality following from the fact that the transformation $\\phi_t^{X}\\circ r_s\\colon M\\to M$ preserves the measure $\\vol$ for any $s,t\\in \\R$.  By dominated convergence,\n\\begin{equation}\n\t\\label{eq:sameaverage}\n\t\\int_{M}\\lim_{t\\to\\infty}k_{f,\\theta}(p,t)\\;\\text{d}\\vol(p)=\\lim_{t\\to\\infty}\\int_{M}k_{f,\\theta}(p,t)\\;\\text{d}\\vol(p)=\\lim_{t\\to\\infty}\\int_{M}f\\;\\text{d}\\vol=\\int_{M}f\\;\\text{d}\\vol\\;.\n\\end{equation}\n\nIn order to finish the proof of the claim, it remains to show that the limit $\\lim_{t\\to\\infty}k_{f,\\theta}(p,t)$ does not depend on $p$. Choose a countable orthonormal basis $(u_{k})_{k\\in I}$ of $L^{2}(M)$ consisting of smooth eigenfunctions of the operator $\\Theta$ and containing a constant function $u_{k_0}$. If $\\Theta u_k=\\frac{i}{2}n_ku_k$ for $n_k\\in \\Z$, then $u_k\\circ r_{s}=e^{2\\pi i n_k s}u_k$ for every $s\\in \\R$ (cf.~Section~\\ref{sec:unitaryrepresentations}). Therefore, we can compute for every $k \\in I$ the $L^{2}$-inner product\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\int_{M}\\lim_{t\\to\\infty}k_{f,\\theta}(p,t)\\;\\overline{u_k(p)}\\;\\text{d}\\vol(p)&=\\lim_{t\\to\\infty}\\int_{M}\\biggl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s\\biggr) \\overline{u_k(p)}\\;\\text{d}p\\\\\n\t\t&=\\lim_{t\\to\\infty}\\frac{1}{\\theta}\\int_{0}^{\\theta}\\int_{M}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\overline{u_k(p)}\\;\\text{d}p\\;\\text{d}s\\\\\n\t\t&=\\lim_{t\\to\\infty}\\frac{1}{\\theta}\\int_0^{\\theta}\\int_{M}f\\circ\\phi_t^{X}(p)\\;\\overline{u_{k}}\\circ r_{-s}(p)\\;\\text{d}p \\;\\text{d}s\\\\\n\t\t&=\\lim_{t\\to\\infty}\\frac{1}{\\theta}\\int_0^{\\theta}e^{-2\\pi i n_k s}\\int_{M}f\\circ\\phi_t^{X}(p)\\;\\overline{u_{k}(p)}\\;\\text{d}p \\;\\text{d}s\\\\\n\t\t&=\\frac{1}{\\theta}\\int_0^{\\theta}e^{-2\\pi i n_ks}\\text{d}s\\cdot \\lim_{t\\to\\infty}\\langle f\\circ \\phi_t^{X},u_k\\rangle\\;,\n\t\\end{split}\n\\end{equation*}\nwhere we used, in successive order, the dominated convergence theorem, Fubini's theorem, invariance of the measure $\\vol$ under the transformation $r_{-s}$ and the definining property of $u_k$. Mixing of the geodesic flow $(\\phi_t^{X})_{t\\in \\R}$ on $M$ (cf.~\\cite[Cor.~2.3]{Bekka-Mayer}) delivers\n\\begin{equation*}\n\t\\lim_{t\\to\\infty}\\langle f\\circ \\phi_t^{X},u_k\\rangle =\\int_{M}f\\;\\text{d}\\vol\\int_{M}u_k\\;\\text{d}\\vol\n\t\\;.\n\\end{equation*}\n\nAs $u_k$ is orthogonal in $L^{2}(M)$ to the constant $u_{k_0}$ for any $k\\neq k_0$, the last expression vanishes for any such $k$. Therefore, we have just shown that the function $p\\mapsto \\lim_{t\\to\\infty}k_{f,\\theta}(p,t)$ is orthogonal to $u_k$ for every $k\\neq k_0$; necessarily, it must be constant. \n\n\\smallskip \n\nDefine now\n\\begin{equation*}\n\t\\mathcal{R}_{\\theta,0,n}f(p,t)=a_{\\theta,0}^{+}f(p)e^{-t}-\\int_t^{\\infty}e^{-\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\;;\n\\end{equation*}\nthen~\\eqref{eq:Gtheta} and~\\eqref{eq:aestnquarter} give\n\\begin{equation*}\n\t|R_{\\theta,0,n}f(p,t)|\\leq 2e \\norm{f}_{\\mathscr{C}^{1}}e^{-t}+\\frac{\\kappa_0}{\\theta}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\int_t^{\\infty}e^{-\\xi}\\;\\text{d}\\xi=\\frac{1}{\\theta}\\bigl(8e\\pi+\\kappa_0(n^{2}+1)\\bigr)\\norm{f}_{\\mathscr{C}^{1}}e^{-t}\n\\end{equation*}\nand, combining~\\eqref{eq:kzero} with~\\eqref{eq:constanttermzero}, we obtain\n\\begin{equation*}\n\tk_{f,\\theta}(p,t)=\\int_{M}f\\;\\text{d}\\vol+e^{-t}\\int_1^{t}-G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi+\\mathcal{R}_{\\theta,0,n}f(p,t)\\;,\n\\end{equation*} \nwhich establishes Theorem~\\ref{thm:case_Theta_eigenfn}(4).\n\n\\subsection{The case $\\mu<0$}\nWe now turn to the case $\\mu<0$ or, equivalently, $\\nu>1$. The solution to~\\eqref{eq:ode} given in~\\eqref{eq:solutionnotquarter} becomes\n\\begin{equation}\n\t\\label{eq:kbelowzero}\n\t\\begin{split}\n\t\tk_{f,\\theta}(p,t)=& e^{\\frac{\\nu-1}{2}t}\\biggl(a^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr) \\\\\n\t\t&+e^{-\\frac{\\nu+1}{2}t}\\biggl(a^{+}_{\\theta,\\mu}f(p)-\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{\\nu-1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\n\nIt follows that the quantity\n\\begin{equation*}\n\t\\begin{split}\n\t\te^{\\frac{\\nu-1}{2}t}\\biggl(a^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)=&-e^{-\\frac{\\nu+1}{2}t}\\biggl(a^{+}_{\\theta,\\mu}f(p)-\\frac{1}{\\nu}\\int_1^{t}e^{-\\frac{\\nu-1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr)\\\\\n\t\t&-k_{f,\\theta}(p,t)\n\t\t+\\frac{e^{\\frac{\\nu-1}{2}t}}{\\nu}\\int_{t}^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\t\\end{split}\n\\end{equation*}\nis uniformly bounded in $t$, which forces \n\\begin{equation*}\n\ta^{-}_{\\theta,\\mu}f(p)+\\frac{1}{\\nu}\\int_1^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi=0 \n\\end{equation*}\nfor every $p \\in M$. Therefore~\\eqref{eq:kbelowzero} results in\n\\begin{equation*}\n\tk_{f,\\theta}(p,t)=e^{-t}\\biggl(-\\frac{e^{\\frac{\\nu+1}{2}t}}{\\nu}\\int_t^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi +e^{-\\frac{\\nu-1}{2}t}a^+_{\\theta,\\mu}f(p)-\\frac{e^{-\\frac{\\nu-1}{2}t}}{\\nu}\\int_1^{t}e^{\\frac{\\nu-1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi \\biggr)\\;.\n\\end{equation*}\n\nWith the help of~\\eqref{eq:Gtheta} and~\\eqref{eq:aestnquarter}, we may estimate the three summands inside the parentheses. For the first, we have\n\\begin{equation*}\n\t\\biggl|\\frac{e^{\\frac{\\nu+1}{2}t}}{\\nu}\\int_t^{\\infty}e^{-\\frac{\\nu+1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr|\\leq \\frac{2\\kappa_0}{\\theta\\nu(\\nu+1)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;,\n\\end{equation*}\nwhile the second can be bounded as\n\\begin{equation*}\n\t\\bigl|e^{-\\frac{\\nu-1}{2}t}a^{+}_{\\theta,\\mu}f(p)\\bigr|\\leq \\frac{e(3+\\nu)}{2\\nu}\\norm{f}_{\\mathscr{C}^{1}}\\;;\n\\end{equation*}\nlastly,\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\biggl|\\frac{e^{-\\frac{\\nu-1}{2}t}}{\\nu}\\int_1^{t}e^{\\frac{\\nu-1}{2}\\xi}G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\\biggr| &\\leq \\frac{2\\kappa_0}{\\theta\\nu(\\nu-1)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}e^{-\\frac{\\nu-1}{2}t}(e^{\\frac{\\nu-1}{2}t}-e^{\\frac{\\nu-1}{2}}) \\\\\n\t\t& \\leq \\frac{2\\kappa_0}{\\theta\\nu(\\nu-1)}(n^{2}+1)\\norm{f}_{\\mathscr{C}^{1}}\\;.\n\t\\end{split}\n\\end{equation*}\n\nWe conclude that \n\\begin{equation*}\n\t|k_{f,\\theta}(p,t)|\\leq\\frac{1}{\\theta} \\biggl(\\frac{4\\kappa_0}{(\\nu-1)(\\nu+1)}+\\frac{2 e\\pi(3+\\nu)}{\\nu}\\biggr)(n^{2}+1) \\norm{f}_{\\mathscr{C}^{1}}e^{-t}\\;,\n\\end{equation*}\nas desired. \n\n\\smallskip\nThis settles Theorem~\\ref{thm:case_Theta_eigenfn}(5).\n\n\n\n\n\n\n\n\n\n\n\\subsection{Regularity of the coefficients in the asymptotic expansion} \n\\label{sec:regularity}\nWe now turn to the examination of the regularity properties of the coefficients $D^{\\pm}_{\\theta,\\mu,n}f$ appearing in the asymptotic expansion of $k_{f,\\theta}(p,t)$, as in Theorem~\\ref{thm:case_Theta_eigenfn}, for $f\\in \\mathscr{C}^{2}(M)$ satisfying $\\square f=\\mu f $ and $\\Theta f=\\frac{i}{2}nf$ for some $\\mu\\in \\text{Spec}(\\square)\\cap\\R_{>0}$ and $n\\in \\Z$. In so doing, we shall complete the proof of Theorem~\\ref{thm:case_Theta_eigenfn}.\n\nLet us fix $\\theta\\in (0,4\\pi]$ throughout this subsection. As follows readily from the definitions of the coefficients \\eqref{eq:anquarter}, \\eqref{eq:aquarter}, \\eqref{eq:Dplusabovequarter}, \\eqref{eq:Dminusabovequarter}, \\eqref{eq:Dquarter} and \\eqref{eq:Dbelowquarter}, it suffices to analyze the regularity of \n\\begin{equation*}\n\tk_{f,\\theta}(p,1)=\\int_0^{\\theta}f\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s, \\; k'_{f,\\theta}(p,1)=\\int_0^{\\theta}Xf\\circ \\phi^{X}_1\\circ r_s(p)\\;\\text{d}s, \\;\\int_1^{\\infty}g(\\xi)G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi\n\\end{equation*}\nas functions of $p\\in M$, with $g(\\xi)$ being a function of the following forms:\n\\begin{equation}\n\t\\label{eq:g}\n\te^{-\\xi\/2},\\;\\xi e^{-\\xi\/2}, \\;e^{-\\xi\/2}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)},\\;e^{-\\xi\/2}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}\\xi\\biggr)},\\; e^{-\\frac{1\\pm\\nu}{2}\\xi}\\;(0<\\nu<1).\n\\end{equation}\n\nWe start with the following elementary lemma:\n\n\\begin{lem}\n\t\\label{lem:diffunderint}\n\tLet $\\theta$ be a positive real number. If $h\\colon M\\to \\C$ is of class $\\mathscr{C}^{1}$, then the function\n\t\\begin{equation*}\n\t\tp\\mapsto \\int_0^{\\theta}h\\circ r_s(p)\\;\\emph{d}s\\;, \\quad p\\in M\n\t\\end{equation*}\n\tis of class $\\mathscr{C}^{1}$ on $M$.\n\\end{lem}\n\\begin{proof}\n\tFix a point $p_0\\in M$, and let $(\\partial_{x_i})_{i=1,2,3}$ be a local frame of the tangent bundle $TM$ around $p_0$. It suffices to prove that, for each $i=1,2,3$, the partial derivative \n\t\\begin{equation}\n\t\t\\label{eq:partialderivative}\n\t\tp\\mapsto\\partial_{x_i}|_p\\biggl(\\int_0^{\\theta}h\\circ r_s\\;\\text{d}s\\biggr)\n\t\\end{equation}\n\texists and is continuous in an open neighborhood of $p$. Upon passing to local smooth charts for $M$, the classical theorem of differentiation under the integral sign ensures the validity of the formal passage\n\t\\begin{equation*}\n\t\tp\\mapsto\\partial_{x_i}|_p\\biggl(\\int_0^{\\theta}h\\circ r_s\\;\\text{d}s\\biggr)=\\int_0^{\\theta}\\partial_{x_i}|_p(h\\circ r_s)\\;\\text{d}s\n\t\\end{equation*}\n\tprovided that there exists a positive real-valued function $\\varphi$ on $[0,\\theta]$, integrable with respect to the Lebesgue measure, such that $|\\partial_{x_i}|_q(h\\circ r_s)|\\leq \\varphi(s)$ for any $q$ in an open neighborhood of $p$. Notice that, by the dominated convergence theorem. this would yield continuity of the partial derivative in~\\eqref{eq:partialderivative} at the same time. The chain rule for the differential gives $\\mathrm{d}(h\\circ r_s)_q=(\\mathrm{d}h)_{r_s(q)}\\circ (\\mathrm{d}r_s)_q$ for any $q \\in M$ and $s\\in [0,\\theta]$. As follows readily from the explicit expression for $r_s$ in~\\eqref{eq:rotationflow} and direct computations, the operator norm\\footnote{Formally, we would need to specify a Riemannian metric on the compact manifold $M$. For the purposes of the proof however, only boundedness of the relevant quantities matters, so that any such metric would serve our goal (cf.~Remark~\\ref{rmk:equivalentmetrics}(1)).} of the linear operator $(\\mathrm{d}r_s)_q$ is uniformly bounded in $q$ and $s$, as the entries of any Jacobian matrix associated to it only involve finite linear combinations of the sine and cosine functions. Therefore, there exists a constant $C>0$ such that $|\\partial_{x_i}|_q(h\\circ r_s)|\\leq C\\norm{h}_{\\mathscr{C}^{1}}$ for any $s\\in [0,\\theta]$ and any $q$ in the domain of definition of the local frame $(\\partial_{x_i})_{i=1,2,3}$. The conclusion is thus achieved by setting $\\varphi$ to be constantly equal to $C\\norm{h}_{\\mathscr{C}^{1}}$. \n\\end{proof}\n\nAs $(\\phi_t^{X})_{t\\in \\R}$ is a smooth flow on $M$ and $f$ is of class $\\mathscr{C}^{2}$, the functions $f\\circ \\phi_1^{X},\\;Xf\\circ \\phi_1^{X}$ are of class $\\cC^{1}$. By virtue of Lemma~\\ref{lem:diffunderint}, the functions $p\\mapsto \\int_0^{\\theta}f\\circ \\phi_1^{X}\\circ r_s(p)\\;\\text{d}s,\\;p\\mapsto \\int_0^{\\theta}Xf\\circ \\phi_1^{X}\\circ r_s(p)\\;\\text{d}s$ are of class $\\cC^{1}$ on $M$. \n\nTherefore, it remains to deal with $\\int_1^{\\infty}g(\\xi)G_{\\theta,n}f(p,\\xi)\\;\\text{d}\\xi$ as a function of $p \\in M$, $g$ being as in~\\eqref{eq:g}. Expanding out the expression from~\\eqref{eq:constantterm}, we obtain that it equals\n\\begin{equation*}\n\t\\begin{split} \\int_1^{\\infty}&g(\\xi)\\biggl(\\frac{n^{2}e^{-\\xi}}{\\theta(1-e^{-2\\xi})^{2}}\\int_0^{\\theta}f\\circ \\phi^{X}_\\xi\\circ r_s(p)\\text{d}s -\\frac{2e^{-\\xi}}{\\theta(1-e^{-2\\xi})}\\int_0^{\\theta}Xf\\circ \\phi^{X}_{\\xi}\\circ r_s(p)\\text{d}s\\\\\n\t\t&+\\frac{2ine^{-2\\xi}}{\\theta(1-e^{-2\\xi})^{2}}(f\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)-f\\circ \\phi^{X}_{\\xi}(p))+\\frac{2}{\\theta (1-e^{-2\\xi})}(Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)-Uf\\circ \\phi^{X}_{\\xi}(p))\\biggr)\\;\\text{d}\\xi\\;.\n\t\\end{split}\n\\end{equation*}\n\nWe shall know treat the four summands separately. \n\n\\begin{lem}\n\t\\label{lem:regularterms}\n\tIf $g$ takes one of the forms in~\\eqref{eq:g}, the functions \n\t\\begin{align}\n\t\t\\label{eq:firstreg}\n\t\t&p\\mapsto \\int_1^{\\infty}\\frac{g(\\xi)e^{-\\xi}}{(1-e^{-2\\xi})^{2}}\\biggl(\\int_0^{\\theta}f\\circ \\phi_{\\xi}^{X}\\circ r_s(p)\\;\\emph{d}s\\biggr)\\emph{d}\\xi\\;,\\\\\n\t\t\\label{eq:secondreg}\n\t\t&p\\mapsto \\int_1^{\\infty}\\frac{g(\\xi)e^{-\\xi}}{1-e^{-2\\xi}}\\biggl(\\int_0^{\\theta}Xf\\circ \\phi_{\\xi}^{X}\\circ r_s(p)\\;\\emph{d}s\\biggr)\\emph{d}\\xi\\;, \\\\\n\t\t\\label{eq:thirdreg}\n\t\t&p\\mapsto \\int_1^{\\infty}\\frac{g(\\xi)e^{-\\xi}}{(1-e^{-2\\xi})^{2}}\\biggl(f\\circ \\phi_{\\xi}^{X}\\circ r_\\theta(p)-f\\circ \\phi_{\\xi}^{X}(p)\\biggr)\\emph{d}\\xi\n\t\\end{align}\n\tare of class $\\mathscr{C}^{1}$ on $M$.\n\\end{lem}\n\\begin{proof}\n\tThe proof proceeds along the same lines as the proof of Lemma~\\ref{lem:diffunderint}. The crucial point is that, for any point $q\\in M$ and any $\\xi\\geq 1$, the operator norm of the differential $(\\mathrm{d}\\phi_\\xi^{X})_{q}$ doesn't exceed (up to a constant factor depending only on the choice of a Riemannian metric on the tangent bundle $TM$) the quantity $e^{\\xi\/2}$, as direct computations allow to verify starting from the explicit expression of $\\phi_{\\xi}^{X}$ in~\\eqref{eq:geodesicflow}. As a consequence, there exists a constant $C>0$ such that\n\t\\begin{equation*}\n\t\t\\biggl|\\frac{g(\\xi)e^{-\\xi}}{(1-e^{-2\\xi})^{2}}\\biggl(\\int_0^{\\theta}\\partial_{x_i}|_q(f\\circ \\phi^{X}_{\\xi}\\circ r_s)\\;\\text{d}s\\biggr)\\biggr|\\leq C\\frac{|g(\\xi)|e^{-\\xi}}{(1-e^{-2\\xi})^{2}}\\int_0^{\\theta}\\norm{f}_{\\mathscr{C}^{1}}e^{\\frac{\\xi}{2}}\\;\\text{d}s=C\\norm{f}_{\\mathscr{C}^{1}}\\frac{|g(\\xi)|e^{-\\frac{\\xi}{2}}}{(1-e^{-2\\xi})^{2}}\n\t\\end{equation*}\n\tfor every $\\xi\\geq 1$ and $i=1,2,3$, where $(\\partial_{x_i})_{i=1,2,3}$ is a local frame of $TM$ around a given fixed point $p_0 \\in M$. Since the function $|g(\\xi)|e^{-\\xi\/2}\/(1-e^{-2\\xi})^{2}$ is integrable on the half-line $[1,\\infty)$, we deduce as in the proof of Lemma~\\ref{lem:diffunderint} that the function in~\\eqref{eq:firstreg} is of class $\\cC^{1}$ on $M$.\n\t\n\tThe same assertion for the remaining two functions in~\\eqref{eq:secondreg} and~\\eqref{eq:thirdreg} follows by a similar argument.  \n\\end{proof}\n\nWhat is left to investigate, up to multiplicative constants, is thus  the regularity of the function\n\\begin{equation}\n\t\\label{eq:holdercontinuousterm}\n\tp\\mapsto \\int_1^{\\infty}\\frac{g(\\xi)}{1-e^{-2\\xi}}\\bigl(Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)-Uf\\circ \\phi^{X}_{\\xi}(p)\\bigr) \\text{d}\\xi\n\\end{equation}\non the manifold $M$. As we shall presently see, the latter depends on the function $g$.\n\n\\begin{lem}\n\t\\label{lem:moreregular}\n\tIf $g(\\xi)=e^{-\\frac{1+\\nu}{2}\\xi}$, then the function in~\\eqref{eq:holdercontinuousterm} is of class $\\mathscr{C}^{1}$ on $M$.\n\\end{lem}\n\\begin{proof}\n\tIt suffices to argue as in the proof of Lemma~\\ref{lem:regularterms} observing that, when $0<\\nu<1$, the function\n\t\\begin{equation*}\n\t\t\\frac{e^{-\\frac{1+\\nu}{2}\\xi}}{1-e^{-2\\xi}}\\;e^{\\frac{\\xi}{2}}=\\frac{e^{-\\frac{\\nu}{2}\\xi}}{1-e^{-2\\xi}}\n\t\\end{equation*}\n\tis integrable on the half-line $[1,\\infty)$.\n\\end{proof}\n\nIt is straightforward to realize that the same argument does not carry over to the other possible forms of $g(\\xi)$ listed in~\\eqref{eq:g}. For those remaining cases, we instead establish H\\\"{o}lder-continuity of the function in~\\eqref{eq:holdercontinuousterm} by a different argument.\n\nFix a Riemannian metric $g$ on the connected manifold $M$, inducing a Riemannian distance function $d$. The choice is immaterial for our purposes, as pointed out in Remark~\\ref{rmk:equivalentmetrics}. We start with the following well-known properties of the flows $(\\phi_t^{X})_{t\\in \\R},\\;(r_{s})_{s\\in \\R}$.\n\n\\begin{lem}\n\t\\label{lem:distanceflows}\n\tThere exist real constants $C_{X,d},C_{\\Theta,d}$, depending only on $d$, such that, for any pair of points $p,q\\in M$, it holds \n\t\\begin{equation}\n\t\t\\label{eq:geodesicdistance}\n\t\td(\\phi_t^{X}(p),\\phi^{X}_t(q))\\leq C_{X,d}\\;e^{|t|}d(p,q)\n\t\\end{equation}\n\tfor every $t\\in \\R$ and\n\t\\begin{equation}\n\t\t\\label{eq:rotationdistance}\n\t\td(r_{s}(p),r_s(q))\\leq C_{\\Theta,d}\\;d(p,q) \n\t\\end{equation}\n\tfor every $s\\in \\R$.\n\\end{lem}\n\\begin{proof}\n\tBy compactness of $M$, we have the freedom to prove the lemma for a judicious choice of $d$. To profit most from the algebraic description of the flows $(\\phi_t)_{t\\in \\R}$ and $(r_{s})_{s\\in \\R}$, we fix a left-invariant Riemannian metric $g_{\\SL_2(\\R)}$ on the Lie group $\\SL_2(\\R)$ and let $d$ be the Riemannian distance determined by the unique Riemannian metric $g$ on $M$ for which the projection map $(\\SL_2(\\R),g_{\\SL_2(\\R)})\\to (M,g)$ is a Riemannian submersion (cf.~\\cite[Cor.~2.29]{Lee-Riemannian}) or, equivalently for a covering map, a local isometry. As $M$ is compact, we can choose a finite open cover $(\\tilde{U_{i}})_{i\\in I}$ of $M$ and a collection $\\mathcal{U}=(U_i)_{i\\in I}$ of open subsets of $\\SL_2(\\R)$ such that, for any $i \\in I$, the restriction of the projection to $U_i$ is an isometry from $U_i$ onto $\\tilde{U_i}$. \n\t\n\tThe distance $d_{\\SL_2(\\R)}$ induced by $g_{\\SL_2(\\R)}$ is locally equivalent to the distance induced by the operator norm $\\norm{\\cdot}_{\\text{op}}$ on the vector space of $2\\times 2$ real matrices corresponding to the Euclidean norm on $\\R^{2}$ (cf.~\\cite[Lem.~9.12]{Einsiedler-Ward}): for every $g\\in \\SL_2(\\R)$, there exists an open neighborhood  $W_g$ of $g$ and a constant $C_{d,g}$ such that \n\t\\begin{equation*}\n\t\tC_{d,g}^{-1}\\norm{x-y}_{\\text{op}}\\leq d_{\\SL_2(\\R)}(x,y)\\leq C_{d,g}\\norm{x-y}_{\\text{op}}\n\t\\end{equation*}\n\tfor any $x,y\\in W_g$. Upon restricting the $U_i$'s (and the $\\tilde{U_i}$'s) if necessary, we may assume that each $\\tilde{U_i}$ is contained in $W_{g_i}$ for some $g_i \\in \\SL_2(\\R)$; define $C_{d}$ to be the supremum of the $C_{d,g_i},\\;i \\in I$. We also select a second finite open cover $\\mathcal{V}=(V_j)_{j\\in J}$ in such a way that the closure of each $V_j$ is compact and contained in some $U_{i(j)}$. Observe that, for every $j \\in J$, the function $\\tau_j\\colon \\overline{V_j}\\to (0,\\infty]$ defined as the first exit time\n\t\\begin{equation*}\n\t\t\\tau_j(p)=\\inf\\{t>0:\\phi_t^{X}(p)\\notin U_{i(j)} \\}\\;, \\quad  p\\in V_{j}\n\t\\end{equation*}\n\tis continuous, and as such attains a strictly positive minimal value $t_{j}$. Set $t_0=\\inf_{j\\in J}t_j$ and let $\\delta_{\\mathcal{V}}$ be a Lebesgue number for the covering $\\mathcal{V}$ (cf.~\\cite[Lem.~27.5]{Munkres}).\n\t\n\tConsider now two points $p,q\\in M$, and suppose first that $d(p,q)\\geq\\delta_{\\mathcal{V}}$; then \n\t\\begin{equation}\n\t\t\\label{eq:largedistance}\n\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))\\leq \\text{diam}_d(M)\\leq \\delta_{\\mathcal{V}}^{-1}\\text{diam}_{d}(M)d(p,q)\\leq \\delta_{\\mathcal{V}}^{-1}\\text{diam}_{d}(M)e^{|t|}d(p,q)\n\t\\end{equation}\n\tfor every $t\\in \\R$, where $\\text{diam}_{d}(M)=\\sup_{p',q'\\in M}d(p,q)$ is the diameter of $M$ with respect to the distance $d$.\n\t\n\tNow assume that $d(p,q)<\\delta_{\\mathcal{V}}$ so that $p$ and $q$ both lie in some $V_j$. Choose representatives $x,y$ of $p,q$, respectively, inside $\\tilde{U}_{i(j)}$; then, for every $0\\leq t<t_0$, we have\n\t\\begin{equation}\n\t\t\\label{eq:smalldistance}\n\t\t\\begin{split}\n\t\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))&\\leq d_{\\SL_2(\\R)}(x\\exp{tX},y\\exp{tX})\\leq C_{d}\\norm{x\\exp{tX}-y\\exp{tX}}_{\\text{op}}\\\\\n\t\t\t&\\leq C_{d}\\norm{x-y}_{\\text{op}}\\norm{\\exp{tX}}_{\\text{op}}\\leq C_{d}^{2}\\;d_{\\SL_2(\\R)}(x,y) e^{|t|\/2}\\\\\n\t\t\t&\\leq C_d^{2}e^{t}d_{\\SL_2(\\R)}(x,y)=C_d^{2}e^{t}d(p,q)\\;,\n\t\t\\end{split}\n\t\\end{equation}\n\twhere we used the fact that $p,q,\\phi_t^{X}(p),\\phi_t^{X}(q)$ all belong to $U_{i}^{(j)}$. If now $t_0\\leq t<2 t_0$, then we distinguishes two cases.\n\t\\begin{itemize}\n\t\t\\item If $d(\\phi_{t_0}^{X}(p),\\phi_{t_0}^{X}(q))\\geq \\delta_{\\mathcal{V}}$, then~\\eqref{eq:largedistance} applies giving\n\t\t\\begin{equation*}\n\t\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))\\leq \\delta_{\\mathcal{V}}^{-1}\\text{diam}_{d}(M)e^{t-t_0}d(\\phi_{t_0}^{X}(p),\\phi_{t_0}^{X}(q))\\leq C_{d}^{2}\\delta_{\\mathcal{V}}^{-1}\\text{diam}_d(M)e^{t}d(p,q)\\;.\n\t\t\\end{equation*}\n\t\tThis estimate is actually valid for any $t\\geq t_0$.\n\t\t\\item If $d(\\phi_{t_0}^{X}(p),\\phi_{t_0}^{X}(q))< \\delta_{\\mathcal{V}}$, then the computations in~\\eqref{eq:smalldistance} are valid for the given $t$ and yield\n\t\t\\begin{equation*}\n\t\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))\\leq C_d^{2}e^{t}d(p,q)\\;.\n\t\t\\end{equation*} \n\t\\end{itemize}\n\tSubdividing the half-line $\\R_{\\geq 0}$ into the intervals $[kt_0,(k+1)t_0),\\;k\\in \\N$, and arguing as above on each of them, we conclude that\n\t\\begin{equation*}\n\t\td(\\phi_t^{X}(p),\\phi_t^{X}(q))\\leq \\sup\\{ C_d^{2}, C_d^{2}\\delta_{\\mathcal{V}}^{-1}\\text{diam}_d(M)\\}e^{t}d(p,q)\n\t\\end{equation*}\n\tfor any $t>0$. \n\t\n\tThe same analysis can be performed, with the appropriate modifications, for times $t<0$. This shows~\\eqref{eq:geodesicdistance}.\n\t\n\tThe inequality in~\\eqref{eq:rotationdistance} is taken care of in an entirely analogous fashion, observing that $\\norm{\\exp(s\\Theta)}_{\\text{op}}=1$ for every $s\\in \\R$.\n\\end{proof}\n\n\nLet us now fix two points $p,q \\in M$. As in the previous proof, we denote by $\\text{diam}_d(M)$ the diameter of $M$; with\n\\begin{equation*} \\text{Lip}_d(Uf)=\\sup\\limits_{p',q'\\in M,\\; p'\\neq q'}\\frac{|Uf(p')-Uf(q')|}{d(p,q)}\n\\end{equation*}\nwe indicate the Lipschitz constant of the function $Uf$ with respect to the distance $d$.\n\nWe may then estimate\n\\begin{equation}\n\t\\label{eq:Holdercontinuity}\n\t\\begin{split}\n\t\t&\\biggl| \\int_1^{\\infty}\\frac{g(\\xi)}{1-e^{-2\\xi}}\\bigl(Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)-Uf\\circ \\phi^{X}_{\\xi}(p)\\bigr) \\text{d}\\xi - \\int_1^{\\infty}\\frac{g(\\xi)}{1-e^{-2\\xi}}\\bigl(Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(q)-Uf\\circ \\phi^{X}_{\\xi}(q)\\bigr) \\text{d}\\xi \\biggr|\\\\\n\t\t&\\leq \\int_1^{\\infty}\\frac{|g(\\xi)|}{1-e^{-2\\xi}}\\bigl(|Uf\\circ \\phi_{\\xi}^{X}\\circ r_{\\theta}(p)-Uf\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(q)|+|Uf\\circ \\phi^{X}_{\\xi}(p)-Uf\\circ \\phi^X_{\\xi}(q)|\\bigr)\\text{d}\\xi\\\\\n\t\t&\\leq (1-e^{-2})^{-1}\\text{Lip}(Uf) \\int_1^{\\infty}|g(\\xi)|\\bigl(\\min\\{\\text{diam}_d(M),C_{X,d}\\;e^{\\xi}d(r_{\\theta}(p),r_{\\theta}(q) \\}+\\\\ &\\quad +\\inf\\{\\text{diam}_d(M),C_{X,d}\\;e^{\\xi}d(p,q) \\}\\bigr)\\text{d}\\xi\\\\\n\t\t&\\leq 2(1-e^{-2})^{-1}\\text{Lip}(Uf)\\int_1^{\\infty}|g(\\xi)|\\inf\\{\\text{diam}_d(M),Cd(p,q)e^{\\xi} \\}\\text{d}\\xi\\;,\n\t\\end{split}\n\\end{equation}\nwhere $C=C_{X,d}\\sup\\{1,C_{\\Theta,d} \\}$.\n\n\\smallskip\nThe following elementary estimates allow to finalize the argument.\n\\begin{lem}\n\t\\label{lem:integralbound}\n\tLet $r,K\\in \\R_{>0},\\;a\\in (0,1)$. Then \n\t\\begin{equation}\n\t\t\\label{eq:firstboundintegral}\n\t\t\\int_1^{\\infty}e^{-a\\xi}\\inf\\{K,re^{\\xi} \\} \\emph{d}\\xi\\leq \\frac{1}{a(1-a)K^{a-1}}\\;r^{a}\\;.\n\t\\end{equation}\n\tFurthermore,\n\t\\begin{equation}\n\t\t\\label{eq:secondboundintegral}\n\t\t\\int_1^{\\infty}\\xi e^{-\\frac{\\xi}{2}}\\inf\\{K,re^{\\xi} \\}\\emph{d}\\xi= 4\\sqrt{K}r^{\\frac{1}{2}}(\\log{k}+\\log{r}^{-1})\\;.\n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\tIt suffices to split the integral as\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\int_1^{\\infty}e^{-a\\xi}\\inf\\{K,re^{\\xi} \\} \\text{d}\\xi&=\\int_1^{\\log{\\frac{K}{r}}}e^{-a\\xi}\\cdot re^{\\xi} \\;\\text{d}\\xi+\\int_{\\log{\\frac{K}{r}}}^{\\infty}e^{-a\\xi}\\cdot K  \\;\\text{d}\\xi\\\\\n\t\t\t&=\\frac{r}{1-a}\\biggl(\\biggr(\\frac{r}{K}\\biggr)^{a-1}-e^{1-a}\\biggr)+\\frac{K}{a}\\biggl(\\frac{r}{K}\\biggr)^{a}\\\\\n\t\t\t&\\leq \\frac{r^{a}}{(1-a)K^{a-1}}+\\frac{r^{a}}{aK^{a-1}}= \\frac{1}{a(1-a)K^{a-1}}\\;r^{a} \\;,\n\t\t\\end{split}\n\t\\end{equation*}\n\twhich delivers the inequality in~\\eqref{eq:firstboundintegral}. Analogous computations allow to establish~\\eqref{eq:secondboundintegral}.\n\\end{proof}\n\nCombining Lemmata~\\ref{lem:regularterms},~\\ref{lem:moreregular},~\\ref{lem:integralbound} together with the estimate in~\\eqref{eq:Holdercontinuity} and the explicit expressions for the coefficients $D^{\\pm}_{\\theta,\\mu,n}f$ in~\\eqref{eq:Dplusabovequarter},~\\eqref{eq:Dminusabovequarter},~\\eqref{eq:Dquarter},~\\eqref{eq:Dbelowquarter} we deduce that:\n\\begin{itemize}\n\t\\item when $\\mu>1\/4$, $D^{\\pm}_{\\theta.\\mu,n}f$ are H\\\"{o}lder continuous with H\\\"{o}lder exponent $1\/2$;\n\t\\item when $\\mu=1\/4$, $D^{+}_{\\theta,1\/4,n}f$ and $D^{-}_{\\theta,1\/4,n}$ are H\\\"{o}lder continuous, the latter with H\\\"{o}lder exponent $1\/2$, while the former with H\\\"{o}lder exponent $1\/2-\\varepsilon$ for every $\\varepsilon>0$;\n\t\\item when $0<\\mu<1\/4$, $D^{+}_{\\theta,\\mu,n}f$ is H\\\"{o}lder continuous with H\\\"{o}lder exponent $\\frac{1-\\nu}{2}$, while $D^{-}_{\\theta,\\mu,n}f$ is of class $\\cC^{1}$ on $M$.\n\\end{itemize}\n\nThe proof of Theorem~\\ref{thm:case_Theta_eigenfn} is achieved.\n\n\\section{Asymptotics for arbitrary functions}\n\\label{sec:arbitraryfn}\n\nThe bulk of this section is devoted to the deduction of Theorem~\\ref{thm:mainexpandingtranslates}, which addresses the asymptotic equidistribution rate of sufficiently regular observables on $M$ not subject to any eigenvalue condition, from the special case of joint eigenfunctions of $\\square$ and $\\Theta$ phrased in Theorem~\\ref{thm:case_Theta_eigenfn}. The argument is crucially based upon the orthogonal decompositions of Sobolev spaces into joint eigenspaces of $\\square$ and $\\Theta$, which is recalled in detail in Section~\\ref{sec:unitaryrepresentations}. We then proceed by proving Theorem~\\ref{thm:mainarbitrarytranslates}, concerning the asymptotic behaviour of arbitrary translates of compact orbits inside $M$; in light of the classical Cartan decomposition of the Lie group $\\SL_2(\\R)$, the result follows from Theorem~\\ref{thm:mainexpandingtranslates} in a fairly straightforward manner. Along the way, we shall also clarify the steps needed to derive, from those two main results, Corollaries~\\ref{cor:effective} and~\\ref{cor:shrinkingarcs}.\n\n\\subsection{Sum estimates on Sobolev norms of eigenfunctions}\n\\label{sec:sumestimates}\n\nBefore proceeding with the proof of Theorem~\\ref{thm:mainexpandingtranslates}, we collect in this subsection a few  estimates relating sums of norms of Sobolev eigenfunctions with a higher-order Sobolev norm of the sum of such functions, which will prove to be instrumental in the sequel. The rationale for those is the fact that the Hilbert-sum decompositions in Section~\\ref{sec:unitaryrepresentations} only provide, by Bessel's inequality (cf.~\\cite[2,~XXIII,~6;~14]{Schwartz}), estimates on the sum of squares of the components' norms, while our approach necessitates bounds on the $\\ell^{1}$-norm (see Section~\\ref{sec:proofexpandingtranslates}).   \n\nNotation is as in Section~\\ref{sec:unitaryrepresentations}. \n\n\\begin{lem}\n\t\\label{lem:finitespectralconstant}\n\tLet $k$ be a natural number. \n\t\\begin{enumerate}\n\t\t\\item The infinite series \n\t\t\\begin{equation}\n\t\t\t\\label{eq:serieseigenvalues}\n\t\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)}\\sum_{n\\in I(\\mu)} \\frac{1}{(1+\\mu+\\frac{n^2}{2})^{k}}\n\t\t\\end{equation}\n\t\tis summable if and only if $k\\geq 2$.\n\t\t\\item The infinite series \n\t\t\\begin{equation}\n\t\t\t\\label{eq:serieseigenvaluessecond}\n\t\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)}\\sum_{n\\in I(\\mu)} \\frac{(n^{2}+1)^{2}}{(1+\\mu+\\frac{n^2}{2})^{k}}\n\t\t\\end{equation}\n\t\tis summable if and only if $k\\geq 3$.\n\t\\end{enumerate}\n\\end{lem}\n\nObserve that the series in~\\eqref{eq:serieseigenvalues} and~\\eqref{eq:serieseigenvaluessecond} consist of nonnegative real numbers: see Lemma~\\ref{lem:diffSobolevnorms}.\n\\begin{proof}\n\tIt is convenient to examine separately the convergence properties of \n\t\\begin{equation}\n\t\t\\label{eq:negativeandpositive}\n\t\t\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{\\geq 0}}\\sum_{n\\in I(\\mu)} \\frac{1}{(1+\\mu+\\frac{n^2}{2})^{k}}\\;\\text{  and  } \\;\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{<0}}\\sum_{n\\in I(\\mu)} \\frac{1}{(1+\\mu+\\frac{n^2}{2})^{k}}\\;.\n\t\\end{equation}\n\tWe know (cf.~Section~\\ref{sec:unitaryrepresentations}) that negative eigenvalues of the Casimir operator are of the form $\\mu_m=-m(m+2)\/4$ for $m$ ranging over the set of positive natural numbers; therefore\n\t\\begin{equation}\n\t\t\\label{eq:negativecasimir}\n\t\t\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{<0}}\\sum_{n\\in I(\\mu)} \\frac{1}{(1+\\mu+\\frac{n^2}{2})^{k}}=\\sum_{m \\in \\N^{\\ast}}\\sum_{n\\in I(\\mu_m)} \\frac{1}{(1-\\frac{m(m+2)}{4}+\\frac{n^2}{2})^{k}}\\;,\n\t\\end{equation}\n\twhich has the same convergence properties of the series\n\t\\begin{equation*}\n\t\t\\sum_{(m,n)\\in \\Z^{2}}\\frac{1}{(1+m^2+n^{2})^{k}}\\;.\n\t\\end{equation*}\n\tBy comparison with the integral\n\t\\begin{equation*}\n\t\t\\int_{\\R^{2}}\\frac{1}{(1+x^2+y^2)^{k}}\\;\\text{d}x\\text{d}y\\;,\n\t\\end{equation*}\n\twhich is convergent if and only if $k\\geq 2$, as it is well-known, we infer that:\n\t\\begin{enumerate}\n\t\t\\item the series in~\\eqref{eq:serieseigenvalues} cannot converge if $k=1$;\n\t\t\\item the series in~\\eqref{eq:negativecasimir} converges for any $k\\geq 2$.\n\t\\end{enumerate}\n\tAs to the first summation in~\\eqref{eq:negativeandpositive}, we may now suppose $k\\geq 2$ and appeal to the Weyl law  for the positive eigenvalues of the Casimir operator (see Theorem~\\ref{thm:Weyllaw}), which we list in increasing order as $\\mu^{(p)}_0=0<\\mu_1^{(p)}<\\cdots<\\mu^{(p)}_{m}<\\cdots$, without multiplicity. Recall that $\\text{area}(S)$ is the volume of the surface $S=\\Gamma\\bsl \\Hyp$ with respect to the hyperbolic area measure. Choose a real number $c>\\text{area}(S)\/4\\pi$; then, there exists $R_0\\in \\R_{>0}$ such that $\\mu^{(p)}_{m}>m\/c$ for any integer $m>cR_0$. On the one hand, the quantity\n\t\\begin{equation*}\n\t\t\\sum_{0\\leq m\\leq cR_0}\\sum_{n\\in I(\\mu^{(p)}_{m})}\\frac{1}{(1+\\mu_m^{(p)}+\\frac{n^2}{2})^{k}}\n\t\\end{equation*}\n\tis a finite sum of converging series. On the other hand, \n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\sum_{m>cR_0}\\sum_{n\\in I(\\mu^{(p)}_{m})}\\frac{1}{(1+\\mu_m^{(p)}+\\frac{n^2}{2})^{k}}&<\\sum_{m>cR_0}\\sum_{n\\in I(\\mu^{(p)}_{m})}\\frac{1}{(1+\\frac{m}{c}+\\frac{n^2}{2})^{k}}\\\\\n\t\t\t&\\leq \\sum_{m>cR_0}\\frac{1}{(1+\\frac{m}{c})^{k}}+\\sum_{m>cR_0}\\sum_{n\\in I(\\mu^{(p)}_{m})\\setminus\\{0\\}}\\frac{1}{(1+\\frac{m}{c}+\\frac{n^2}{2})^{k}}\\;.\n\t\t\\end{split}\n\t\\end{equation*}\n\tThe first summand in the last expression converges obviously whenever $k\\geq 2$, and so does the second by comparison with the integral\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\int_{cR_0}^{\\infty}\\int_0^{\\infty}\\frac{1}{\\bigl(1+\\frac{x}{c}+\\frac{y^2}{2}\\bigr)^{k}}\\;\\text{d}y\\;\\text{d}x&=\\int_{cR_0}^{\\infty}\\frac{1}{\\bigl(1+\\frac{x}{c}\\bigr)^{k}}\\int_0^{\\infty}\\frac{\\sqrt{2}\\bigl(1+\\frac{x}{c}\\bigr)^{1\/2}}{(1+u^{2})^{k}}\\;\\text{d}u\\;\\text{d}x\\\\\n\t\t\t&=\\biggl(\\int_{cR_0}^{\\infty}\\frac{\\sqrt{2}}{\\bigl(1+\\frac{x}{c}\\bigr)^{k-1\/2}}\\;\\text{d}x\\biggr)\\biggl(\\int_0^{\\infty}\\frac{1}{(1+u^{2})^{k}}\\;\\text{d}u\\biggr)<\\infty\\;.\n\t\t\\end{split}\n\t\\end{equation*} \n\tAs to the assertion for the series in~\\eqref{eq:serieseigenvaluessecond}, it follows readily by running the argument above with the appropriate modifications.\n\\end{proof}\n\nLeveraging the estimates in Lemma~\\ref{lem:finitespectralconstant}, we are now in a position to show:\n\n\\begin{prop}\n\t\\label{prop:L1bound}\n\tLet $k\\geq 2$ be an integer. There exists a constant $C_{\\emph{Spec},k}>0$ depending only on $k$ and on the spectrum of the Laplace-Beltrami operator on the hyperbolic surface $S$, such that the following holds. Let $s$ be a positive real number, $f$ a function in $W^{s+k}(H)$; for any $\\mu \\in \\emph{Spec}(\\square)$ and $n\\in I(\\mu)$, let $f_{\\mu,n}$ be the orthogonal projection of $f$, with respect to the inner product in $W^{s+k}(H)$, onto the closed subspace $W^{s+k}(H_{\\mu,n})$. Then\n\t\\begin{equation}\n\t\t\\label{eq:L1bound}\n\t\t\\sum_{\\mu \\in \\emph{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s}}\\leq C_{\\emph{Spec},k}\\norm{f}_{W^{s+k}}\\;.\n\t\\end{equation}\n\\end{prop}\n\\begin{proof}\n\tRecall from Lemma~\\ref{lem:diffSobolevnorms} that, for any $\\mu \\in \\text{Spec}(\\square)$ and $n\\in I(\\mu)$,\n\t\\begin{equation*}\n\t\t\\norm{f_{\\mu,n}}_{W^{s}}^2=\\biggl(1+\\mu+\\frac{n^2}{2}\\biggr)^{-k}\\norm{f_{\\mu,n}}_{W^{s+k}}^2\\;.\n\t\\end{equation*}\n\tUsing the Cauchy-Schwartz inequality, we get\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s}}&=\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\biggl(1+\\mu+\\frac{n^2}{2}\\biggr)^{-k\/2}\\norm{f_{\\mu,n}}_{W^{s+k}}\\\\\n\t\t\t&\\leq \\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\frac{1}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{k}}\\biggr)^{1\/2}\\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s+k}}^{2}\\biggr)^{1\/2}\\;.\n\t\t\\end{split}\n\t\\end{equation*}\n\tThe inequality in~\\eqref{eq:L1bound} is thus a consequence of Parseval's identity (cf.~\\cite[2,~XXIII,~6;~17]{Schwartz}) \n\t\\begin{equation*}\n\t\t\\norm{f}_{W^{s+k}}^{2}=\\sum_{\\mu\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s+k}}^{2}\\;,\n\t\\end{equation*}\n\twhere we define the constant $C_{\\text{Spec},k}$ as \n\t\\begin{equation*}\n\t\tC_{\\text{Spec},k}=\\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\frac{1}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{k}}\\biggr)^{1\/2}\\;,\n\t\\end{equation*} \n\twhich is finite by Lemma~\\ref{lem:finitespectralconstant} and satisfies the dependence properties claimed in the statement (cf.~Sections~\\ref{sec:unitaryrepresentations},~\\ref{sec:Sobolev}).\n\\end{proof}\n\n\\subsection{Equidistribution of expanding translates}\n\\label{sec:proofexpandingtranslates}\n\nWe are now ready to prove Theorem~\\ref{thm:mainexpandingtranslates}.\nLet $\\theta$ be a real parameter in the interval $(0,4\\pi]$, $s$ a real number satisfying $s>11\/2$, $f$ a function in $W^{s}(M)$. Keeping with the notation introduced in the foregoing subsection, we denote by $f_{\\mu,n}\\in W^{s}(H_{\\mu,n})$ the orthogonal projection of $f$ onto $W^{s}(H_{\\mu,n})$, for any Casimir eigenvalue $\\mu$ and any $n\\in I(\\mu)$. In what follows, the equivalence classes $f$ and $f_{\\mu,n}$ are identified with their unique\\footnote{Uniqueness is a result of the fact that the uniform measure $\\vol$ on $M$ is fully supported.} continuous representatives. The asymptotic expansion in~\\eqref{eq:asymptoticgeneral} will result from the sum of the contributions of each component $f_{\\mu,n}$, which are provided by Theorem~\\ref{thm:case_Theta_eigenfn}; we now expose the details.\n\nChoose a real parameter $s'$ satisfying $3\/2<s'\\leq s-2$;\nthe Sobolev Embedding Theorem (Theorem~\\ref{thm:Sobolevembedding}) gives the bound $\\norm{f_{\\mu,n}}_{\\infty}\\leq C_{0,s'}\\norm{f_{\\mu,n}}_{W^{s'}}$ for any $\\mu$ and $n$ as before, for some constant $C_{0,s'}>0$ depending only on $s'$ and on the manifold $M$.\nFor any point $q \\in M$, we estimate\n\\begin{equation}\n\t\\label{eq:uniformbound}\n\t\\begin{split}\n\t\t\\sum_{\\mu\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}|f_{\\mu,n}(q)|&\\leq \\sum_{n\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{\\infty}\\leq C_{0,s'}\\sum_{\\mu\\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s'}}\\\\\n\t\t\\leq & C_{0,s'}C_{\\text{Spec},2}\\norm{f}_{W^{s'+2}}\\;,\n\t\\end{split}\n\\end{equation}\nthe last inequality being given by Proposition~\\ref{prop:L1bound}. Select now a base point $p \\in M$, which will remain fixed until the end of this subsection. By virtue of~\\eqref{eq:uniformbound}, the dominated convegence theorem for infinite series yields\n\\begin{equation}\n\t\\label{eq:sumintegral}\n\t\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s=\\sum_{\\mu \\in \\text{Spec}(\\square)}\\sum_{n\\in I(\\mu)}\\frac{1}{\\theta}\\int_0^{\\theta}f_{\\mu,n}\\circ \\phi^{X}_t\\circ r_s(p)\\;\\text{d}s\\;.\n\\end{equation}\nObserve that, since $s>11\/2$, the components $f_{\\mu,n}$ are of class $\\cC^{2}$ by the Sobolev Embedding Theorem (Theorem~\\ref{thm:Sobolevembedding}); to each summand on the right-hand side of~\\eqref{eq:sumintegral}, we may thus apply Theorem~\\ref{thm:case_Theta_eigenfn}, which delivers on a formal level the equality\\footnote{Notice that $\\int_{M}f_{\\mu,n}\\;\\text{d}\\vol=0$ for every Casimir eigenvalue $\\mu\\neq 0$ and every $n\\in I(\\mu)$, as $f_{\\mu,n}$ is orthogonal to the joint eigenspace $H_{0,0}$ which contains the constant functions. For the same reason, $\\int_{M}f_{0,n}\\;\\text{d}\\vol=0$ for every $n\\in I(0)\\setminus\\{0\\}$. Therefore, dominated convergence gives $\\int_{M}f_{0.0}\\;\\text{d}\\vol=\\int_{M}f\\;\\text{d}\\vol$.}\n\\begin{equation}\n\t\\label{eq:fullexpansion}\n\t\\begin{split}\n\t\t\\frac{1}{\\theta}&\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s=\\int_{M}f\\;\\text{d}\\vol\\\\ \n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu\\geq 1\/4}e^{-\\frac{t}{2}}\\biggl(\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\biggl(\\sum_{n\\in I(\\mu)}D^{+}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\biggr)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t\\biggr)}\\biggl(\\sum_{n\\in I(\\mu)}D^{-}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\biggr)\\biggr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu< 1\/4}e^{-\\frac{1+\\nu}{2}t}\\biggl(\\sum_{n\\in I(\\mu)}D^{+}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\biggr)+e^{-\\frac{1-\\nu}{2}t}\\biggl(\\sum_{n\\in I(\\mu)}D^{-}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\biggr) \\\\\n\t\t&+ \\varepsilon_0\\biggl(e^{-\\frac{t}{2}}\\biggl(\\sum_{n\\in I(1\/4)}D^{+}_{\\theta,1\/4,n}f_{1\/4,n}(p)\\biggr)+te^{-\\frac{t}{2}}\\biggl(\\sum_{n\\in I(1\/4)}D^{-}_{\\theta,1\/4,n}f_{1\/4,n}(p)\\biggr)\\biggr)\\\\\n\t\t&+ \\mathcal{R}_{\\theta,\\text{pos}}f(p,t)+ e^{-t}\\mathcal{M}_{\\theta,0}f(p,t)+\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,t) +\\mathcal{R}_{\\theta,\\text{d}}f(p,t)\n\t\\end{split}\n\\end{equation}\nfor every $t\\geq 1$,\nwhere $\\varepsilon_0$ is defined in~\\eqref{eq:epszero}, $G_{\\theta,n}f_{0,n}$ is as in~\\eqref{eq:constantterm} and the quantities\n\\begin{equation*}\n \\mathcal{R}_{\\theta,\\text{pos}}f(p,t),\\quad\\mathcal{M}_{\\theta,0}f(p,t),\\quad\\mathcal{R}_{\\theta,\\text{d}}f(p,t)\n\\end{equation*} \n are defined as follows:\n\\begin{align}\n\t\\label{eq:remainderpos}\n\t&\\mathcal{R}_{\\theta,\\text{pos}}f(p,t)=\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>0}\\;\\sum_{n\\in I(\\mu)}\\mathcal{R}_{\\theta,\\mu,n}f(p,t)\\;,\\\\\n\t\\label{eq:remaindernull}\n\t&\\mathcal{M}_{\\theta,0}f(p,t)=\\sum_{n\\in I(0)}\\int_1^{t}-G_{\\theta,n}f_{0,n}(p,\\xi)\\;\\text{d}\\xi \\;,\\\\\n\t\\label{eq:remainderneg}\n\t&\\mathcal{R}_{\\theta,\\text{d}}f(p,t)=\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu<0}\\sum_{n\\in I(\\mu)}\\frac{1}{\\theta}\\int_0^{\\theta}f_{\\mu,n}\\circ\\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s\\;.\n\\end{align}\nThe equality in~\\eqref{eq:asymptoticgeneral} would follow directly from~\\eqref{eq:fullexpansion} by defining\n\\begin{equation}\n\t\\label{eq:defDthetamu}\n\tD^{\\pm}_{\\theta,\\mu}f(p)=\\sum_{n\\in I(\\mu)}D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}(p)\\;, \\quad p\\in M,\\; \\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}\n\\end{equation} \nand\n\\begin{equation}\n\t\\label{eq:remainder}\n\t\\mathcal{R}_{\\theta}f(p,t)=\\mathcal{R}_{\\theta,\\text{pos}}f(p,t)+e^{-t}\\mathcal{M}_{\\theta,0}f(p,t)+\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,t)+\\mathcal{R}_{\\theta,\\text{d}}f(p,t)\\;,\\;p\\in M,\\;t\\geq 1.\n\\end{equation}\nIt is left to show that all the infinite sums we are considering with a formal meaning are actually convergent.\n\nLet us begin by examining the sums in~\\eqref{eq:defDthetamu}. Fix $\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}$; for any $p \\in M$ and $n\\in I(\\mu)$, we have from Theorem~\\ref{thm:case_Theta_eigenfn} that\n\\begin{equation*}\n\t|D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}(p)|\\leq \\norm{D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}}_{\\infty}\\leq \\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f_{\\mu,n}}_{\\mathscr{C}^{1}}\\;.\n\\end{equation*}\nChoose now a real parameter $s''$ so that $5\/2<s''\\leq s-3$; then Theorem~\\ref{thm:Sobolevembedding} allows to deduce\n\\begin{equation}\n\t\\label{eq:uniformboundD}\n\t\\norm{D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}}_{\\infty}\\leq C_{1,s''}\\frac{\\kappa(\\mu)}{\\theta}(n^{2}+1)\\norm{f_{\\mu,n}}_{W^{s''}}\\;.\n\\end{equation}\nNow, in view of Lemma~\\ref{lem:diffSobolevnorms} and applying Cauchy-Schwartz's inequality and Parseval's identity (cf.~\\cite[2,~XXIII,~6;~17]{Schwartz}), we get\n\\begin{equation}\n\t\\label{eq:sameCasimir}\n\t\\begin{split}\n\t\t\\sum_{n\\in I(\\mu)}(n^{2}+1)\\norm{f_{\\mu,n}}_{W^{s''}}&=\\sum_{n\\in I(\\mu)}\\frac{n^{2}+1}{\\bigl(1+\\mu+\\frac{n^2}{2})^{3\/2}}\\norm{f_{\\mu,n}}_{W^{s''+3}}\\\\\n\t\t&\\leq \\biggl(\\sum_{n\\in I(\\mu)}\\frac{(n^2+1)^{2}}{\\bigl(1+\\mu+\\frac{n^2}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\biggl(\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}^{2}_{W^{s''+3}}\\biggr)^{1\/2}\\\\\n\t\t&= \\biggl(\\sum_{n\\in I(\\mu)}\\frac{(n^{2}+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2} \\norm{f_{\\mu}}_{W^{s''+3}}\n\t\\end{split}\n\\end{equation}\nwhere $f_{\\mu}$ is the orthogonal projection of $f$ onto the closed subspace $W^{s}(H_{\\mu})$, and the infinite sum in the last expression converges (see Lemma~\\ref{lem:finitespectralconstant}).\n\nDefining\n\\begin{equation}\n\t\\label{eq:spectralconstant}\n\tC_{\\mu}= \\biggl(\\sum_{n\\in I(\\mu)}\\frac{(n^{2}+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\;,\n\\end{equation}\nour argument thus leads, combining~\\eqref{eq:uniformboundD} and~\\eqref{eq:sameCasimir}, to the estimate\n\\begin{equation}\n\t\\label{eq:sumcoefficients}\n\t\\sum_{n\\in I(\\mu)}\\norm{D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}}_{\\infty}\\leq C_{1,s''}C_{\\mu} \\frac{\\kappa(\\mu)}{\\theta}\\norm{f_{\\mu}}_{W^{s''+3}}\\;,\n\\end{equation}\nwhich implies that the sums $\\sum_{n\\in I(\\mu)}D^{\\pm}_{\\theta,\\mu,n}f_{\\mu,n}$ converge normally in the Banach space $\\mathscr{C}(M)$, hence absolutely and uniformly for all $p \\in M$. In particular, the functions $D^{\\pm}_{\\theta,\\mu}f$ are well-defined, continuous on $M$ and fulfill the upper bound\n\\begin{equation}\n\t\\label{eq:boundDmu}\n\t\\norm{D^{\\pm}_{\\theta,\\mu}f}_{\\infty}\\leq C_{1,s-3}C_{\\mu} \\frac{\\kappa(\\mu)}{\\theta}\\norm{f_{\\mu}}_{W^{s}},\n\\end{equation}\nobtained by picking $s''=s-3$ in~\\eqref{eq:sumcoefficients}. \n\nWe now consider sums over all positive Casimir eigenvalues. We have\n\\begin{equation*}\n\t\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4} |D^{\\pm}_{\\theta,\\mu}f(p)|\\leq \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4} \\norm{D^{\\pm}_{\\theta,\\mu}f}_{\\infty}\\leq \\frac{C_{1,s-3}}{\\theta}\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}C_{\\mu}\\kappa(\\mu)\\norm{f_{\\mu}}_{W^{s}}\\;.\n\\end{equation*}\nObserve that $\\kappa(\\mu)$ is uniformly bounded by a constant $C_{\\text{Spec},\\text{pos}}$ depending only on the infimum of the set $ \\text{Spec}(\\square)\\cap (1\/4,\\infty)$ (see~\\eqref{eq:kappamu}); recalling the definition of $C_{\\mu}$ in~\\eqref{eq:spectralconstant}, we apply once again Cauchy-Schwartz's inequality and Parseval's identity to infer\n\\begin{equation}\n\t\\label{eq:plustwo-three}\n\t\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4} \\norm{D^{\\pm}_{\\theta,\\mu}f}_{\\infty}\\leq C_{\\text{Spec},\\text{pos}}C_{1,s-3}\\norm{f}_{W^{s}}\\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\;\\sum_{n\\in I(\\mu)}\\frac{(n^2+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\;,\n\\end{equation}\nwhere the term between parentheses on the right-hand side is finite because of Lemma~\\ref{lem:finitespectralconstant}. \n\nSince the spectrum of the Casimir operator is discrete, there are only finitely many distinct eigenvalues in the interval $(0,1\/4)$, so that the series \n\\begin{equation*}\n\t\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\norm{D^{\\pm}_{\\theta,\\mu}f}_{\\infty}\n\\end{equation*}\ninvolves only finitely many additional terms with respect to~\\eqref{eq:plustwo-three}; each of those terms can be bounded with the help of~\\eqref{eq:boundDmu}. The claim in~\\eqref{eq:boundDthetamu} follows, by defining the constant $C'_{\\text{Spec}}$ appropriately in terms of $C_{\\text{Spec},\\text{pos}}$ and of the $C_{\\mu}$ for $0<\\mu<1\/4$.\n\n\\medskip\nIn order to finalize the proof of Theorem~\\ref{thm:mainexpandingtranslates}, we address now the remainder terms defined in~\\eqref{eq:remainderpos},\\linebreak\n\\eqref{eq:remaindernull}, ~\\eqref{eq:remainderneg} and~\\eqref{eq:remainder}.\n\nWe start with the term in~\\eqref{eq:remainderpos} stemming from positive Casimir eigenvalues. Define $\\mu_{\\text{princ}}$ to be the infimum of $\\text{Spec}(\\square)\\cap (1\/4,\\infty)$, and let  $\\nu_{\\text{princ}}$ be the corresponding parameter fulfilling $1-\\nu_{\\text{princ}}^{2}=4\\mu_{\\text{princ}}$. Using the bounds for the remainder terms $\\mathcal{R}_{\\theta,\\mu,n}f_{\\mu,n}$ corresponding to the single components $f_{\\mu,n}$, provided by Theorem~\\ref{thm:case_Theta_eigenfn}, we estimate\n\n\n\\begin{equation}\n\t\\label{eq:estimateremainder}\n\t\\begin{split}\n\t\t&\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\;\\sum_{n\\in I(\\mu)}|\\mathcal{R}_{\\theta,\\mu,n}f_{\\mu,n}(p,t)|\\\\\n\t\t&\\leq \\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\frac{8\\kappa_0}{\\theta\\Im{\\nu}}e^{-t}\\sum_{n\\in I(\\mu)}(n^{2}+1)\\norm{f_{\\mu,n}}_{\\cC^{1}}\\\\\n\t\t&\\leq  \\frac{8\\kappa_0C_{1,s-3}}{\\theta}e^{-t} \\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\frac{1}{\\Im{\\nu}}\\sum_{n\\in I(\\mu)}(n^{2}+1)\\norm{f_{\\mu,n}}_{W^{s-3}}\\\\\n\t\t&\\leq  \\frac{8\\kappa_0C_{1,s-3}}{\\theta\\; \\Im{\\nu_{\\text{princ}}}}e^{-t} \\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\biggl(\\sum_{n\\in I(\\mu)}\\frac{(n^{2}+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\biggl(\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s}}^{2}\\biggr)^{1\/2}\\\\\n\t\t&\\leq  \\frac{8\\kappa_0C_{1,s-3}}{\\theta\\; \\Im{\\nu_{\\text{princ}}}}e^{-t} \\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\sum_{n\\in I(\\mu)}\\frac{(n^{2}+1)^2}{\\bigl(1+\\mu+\\frac{n^{2}}{2}\\bigr)^{3}}\\biggr)^{1\/2}\\biggl(\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\sum_{n\\in I(\\mu)}\\norm{f_{\\mu,n}}_{W^{s}}^{2}\\biggr)^{1\/2}\\\\\n\t\t&\\leq \\frac{8\\kappa_0C_{1,s-3}C_{\\text{Spec},3}}{\\theta}\\frac{1}{\\Im{\\nu_{\\text{princ}}}}\\norm{f}_{W^{s}}e^{-t}\n\t\\end{split}\n\\end{equation}\nfor any $t\\geq 1$, applying the bound in~\\eqref{eq:estremainderabovequarter}, Theorem~\\ref{thm:Sobolevembedding}, the Cauchy-Schwartz's inequality (twice) and Bessel's inequality (cf.~\\cite[2,~XXIII,~6;~14]{Schwartz}) to $W^{s}(M)$. \nSimilarly, the bounds in~\\eqref{eq:estquarter} and~\\eqref{eq:estremainderbelowquarter} yield, respectively,\n\\begin{equation*}\n\t\\sum_{n\\in I(1\/4)}|\\mathcal{R}_{\\theta,1\/4,n}f_{1\/4,n}(p,t)|\\leq \\frac{4\\kappa_0 C_{1,s-3}C_{\\text{Spec},3}}{\\theta}\\norm{f}_{W^{s}}(t+1)e^{-t}\n\\end{equation*}\nand\n\\begin{equation*}\n\t\\sum_{\\mu \\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\;\\sum_{n\\in I(\\mu)}|\\mathcal{R}_{\\theta,\\mu,n}f_{\\mu,n}(p,t)|\\leq \\frac{4\\kappa_0 C_{1,s-3}C_{\\text{Spec},3}C_{\\text{comp}}}{\\theta}\\norm{f}_{W^{s}}e^{-t}\n\\end{equation*}\nfor any $t\\geq 1$, where we set \n\\begin{equation*}\n\tC_{\\text{comp}}=\\sum_{\\mu \\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\frac{1}{\\nu(1-\\nu)(1+\\nu)}\\;.\\end{equation*}\nDefining thus\n\\begin{equation}\n\t\\label{eq:Cpos}\n\tC_{\\text{pos}}=\\frac{2}{\\Im{\\nu_{\\text{princ}}}}+C_{\\text{comp}}+1\n\\end{equation}\nand applying the triangular inequality for infinite sums, we get from~\\eqref{eq:remainderpos} that\n\\begin{equation}\n\t\\label{eq:estimateposremainder}\n\t|\\mathcal{R}_{\\theta,\\text{pos}}f(p,t)|\\leq \\frac{4\\kappa_0C_{1,s-3}C_{\\text{Spec},3}C_{\\text{pos}}}{\\theta}\\norm{f}_{W^{s}}(t+1)e^{-t}\\;, \\quad t\\geq 1.\n\\end{equation}\nAn entirely analogous argument, using the bound in~\\eqref{eq:remainderzero}, shows that \n\\begin{equation}\n\t\\label{eq:estimatezeroremainder}\n\t\\biggl|\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,t)\\biggr|\\leq\\frac{ (8e\\pi+\\kappa_0)C_{1,s-3}C_{\\text{Spec},3}}{\\theta}\\norm{f}_{W^{s}}e^{-t}\\;,\\quad t\\geq 1.\n\\end{equation}\nDefine now \n\\begin{equation*}\n\tC_{\\text{disc}}=\\biggl(\\inf\\limits_{\\mu \\in \\text{Spec}(\\square),\\;\\mu<0}|\\mu|\\biggr)^{-1}+\\frac{2e\\pi}{\\kappa_0}\\sup_{\\mu \\in \\text{Spec}(\\square),\\;\\mu<0}\\frac{3+\\nu}{\\nu}\\;.\n\\end{equation*}\nThen, the bound in~\\eqref{eq:estimatediscreteseries} leads to\n\\begin{equation*}\n\t\\begin{split}\n\t\t&\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu<0}\\;\\sum_{n\\in I(\\mu)}\\frac{1}{\\theta}\\biggl|\\int_0^{\\theta}f_{\\mu,n}\\circ \\phi^{X}_{t}\\circ r_{s}(p)\\;\\text{d}s\\biggr| \\\\\n\t\t&\\leq \\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu<0}\\frac{C_{1,s-3}}{\\theta}\\biggl(\\frac{4\\kappa_0}{(\\nu-1)(\\nu+1)}+\\frac{2e\\pi(3+\\nu)}{\\nu}\\biggr)e^{-t}\\sum_{n\\in I(\\mu)}(n^{2}+1)\\norm{f_{\\mu,n}}_{W^{s-3}}\\\\\n\t\t&\\leq \\frac{\\kappa_0C_{1,s-3}C_{\\text{Spec},3}C_{\\text{disc}}}{\\theta}\\norm{f}_{W^{s}}e^{-t}\n\t\\end{split}\n\\end{equation*}\nfor any $t\\geq 1$, arguing as in~\\eqref{eq:estimateremainder}. It follows at once from~\\eqref{eq:remainderneg} that\n\\begin{equation}\n\t\\label{eq:estimatenegremainder}\n\t|\\mathcal{R}_{\\theta,\\text{d}}f(p,t)|\\leq \\frac{\\kappa_0C_{1,s-3}C_{\\text{Spec},3}C_{\\text{disc}}}{\\theta}\\norm{f}_{W^{s}}e^{-t}\n\\end{equation}\nfor any $t\\geq 1$.\nWe finally come to the estimate of the term $\\cM_{\\theta,0}f(p,t)$, defined in~\\eqref{eq:remaindernull}.\nThe inequality in~\\eqref{eq:Gtheta} gives\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\sum_{n\\in I(0)}\\biggl|\\int_1^{t}-G_{\\theta,n}f_{0,n}(p,\\xi)\\;\\text{d}\\xi\\biggr|&\\leq \\frac{\\kappa_0}{\\theta}\\sum_{n\\in I(0)}(n^{2}+1)\\int_1^{t}\\norm{f_{0,n}}_{\\mathscr{C}^{1}}\\;\\text{d}\\xi=\\frac{\\kappa_0}{\\theta}(t-1)\\biggl(\\sum_{n\\in I(0)}(n^{2}+1)\\norm{f_{0,n}}_{\\mathscr{C}^{1}}\\bigg)\\\\\n\t\t&\\leq \\frac{\\kappa_0C_{1,s-3}}{\\theta}(t-1)\\biggl(\\sum_{n\\in I(0)}(n^{2}+1)\\norm{f_{0,n}}_{W^{s}}\\biggr)\\;,\n\t\\end{split}\n\\end{equation*}\nso that \n\\begin{equation}\n\t\\label{eq:estimatezeromain}\n\t|\\mathcal{M}_{\\theta,0}f(p,t)|\\leq \\frac{\\kappa_0C_{1,s-3}C_{\\text{Spec},3}}{\\theta}\\norm{f}_{W^{s}}(t-1)\n\\end{equation}\nfor any $t\\geq 1$.\nRecalling~\\eqref{eq:remainder} and combining the estimates in~\\eqref{eq:estimateposremainder},~\\eqref{eq:estimatezeroremainder},~\\eqref{eq:estimatenegremainder} and~\\eqref{eq:estimatezeromain} we conclude that, for any $t\\geq 1$,\n\\begin{equation*}\n\t\\begin{split}\n\t\t|\\mathcal{R}_{\\theta}f(p,t)|&\\leq |\\mathcal{R}_{\\theta,\\text{pos}}f(p,t)|+e^{-t}|\\mathcal{M}_{\\theta,0}f(p,t)|+\\sum_{n\\in I(0)}|\\mathcal{R}_{0,\\theta,n}f_{0,n}(p,t)|+|\\mathcal{R}_{\\theta,\\text{d}}f(p,t)|\\\\\n\t\t&\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}}{\\theta}\\norm{f}_{W^s}(t+1)e^{-t}\\;,\n\t\\end{split}\n\\end{equation*}\nwhere we set\n\\begin{equation*}\n\tC_{\\text{Spec}}=C_{\\text{Spec},3}\\bigl(8e\\pi+\\kappa_0(2+C_{\\text{pos}}+C_{\\text{disc}})\\bigr)\\;,\n\\end{equation*}\nwhich ostensibly depends only on the spectrum of the Casimir operator.\n\nThe proof of Theorem~\\ref{thm:mainexpandingtranslates} is complete.\n\n\\subsubsection*{Effective equidistribution and shrinking circle arcs}\nIn this paragraph we briefly comment on the proof of Corollaries~\\ref{cor:effective} and~\\ref{cor:shrinkingarcs}.\n\nAs to Corollary~\\ref{cor:effective}, it suffices to define the function $D^{\\text{main}}_{\\theta}f\\colon M\\times \\R_{\\geq 0}\\to \\C$ as follows:\n\\begin{itemize}\n\t\\item $ D^{\\text{main}}_{\\theta}f=D^{-}_{\\theta,\\mu_*}f$\n\tif the spectral gap $\\mu_{*}\\leq 1\/4$;\n\t\\item \n\t$D^{\\text{main}}_{\\theta}f=D^{+}_{\\theta,\\mu_*}f+D^{-}_{\\theta,\\mu_*}f$ if $\\mu_*>1\/4$.\n\t\n\\end{itemize}\n\nThe effective equidistribution statement in~\\eqref{eq:effective} then follows directly from the asymptotics in~\\eqref{eq:asymptoticgeneral}.\n\nNow suppose that we let the boundaries of the parametrization depend on the time $t$, so as to deal with a collection of time-varying subarcs\n\\begin{equation*}\n \\gamma_t=\\{\\phi^{X}_{t}\\circ r_s(p):\\theta_1(t)\\leq s\\leq  \\theta_2(t) \\} \n\\end{equation*} \n as in the statement of Corollary~\\ref{cor:shrinkingarcs}. If, as in the assumptions to the latter, we suppose that $\\theta_2(t)-\\theta_1(t)\\geq \\eta(t)e^{-\\frac{1-\\Re{\\nu_*}}{2}t}$ for every sufficiently large $t$, where $\\nu_*$ corresponds to the spectral gap $\\mu_*$ and $\\eta\\colon \\R_{>0}\\to \\R_{>0}$ satisfies $\\eta(t)\\to\\infty$ as $t\\to\\infty$, then we obtain\n\\begin{equation}\n\t\\label{eq:varyingarc}\n\t\\begin{split}\n\t\t\\frac{1}{\\theta_2(t)-\\theta_1(t)}&\\int_{\\theta_1(t)}^{\\theta_2(t)}f\\circ \\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s = \\frac{1}{\\theta_2(t)-\\theta_1(t)}\\int_{0}^{\\theta_2(t)-\\theta_1(t)}f\\circ \\phi^{X}_t\\circ r_{s+\\theta_1(t)}(p)\\;\\text{d}s\\\\\n\t\t&=\\int_{M}f\\;\\text{d}\\vol+D^{\\text{main}}_{\\theta_2(t) - \\theta_1(t)}f\\left( r_{\\theta_1(t)}(p)\\right)t^{\\varepsilon_0}e^{-\\frac{1-\\Re{\\nu_*}}{2}t}+o(e^{-\\frac{1-\\Re{\\nu_*}}{2}t})\\;\\;.\n\t\\end{split}\n\\end{equation}\nThe bound \\eqref{eq:boundDthetamain} results into \n\\begin{equation*}\n\t|D^{\\text{main}}_{\\theta_2(t) - \\theta_1(t)}f\\left( r_{\\theta_1(t)}(p)\\right)|\\leq C_{1,s-3}C_{\\text{Spec}}'\\norm{f}_{W^{s}}\\biggl(\\frac{1}{\\theta_2(t)-\\theta_1(t)}\\biggr)\\leq  C_{1,s-3}C'_{\\text{Spec}}\\norm{f}_{W^s}  \\eta(t)^{-1}e^{\\frac{1-\\Re{\\nu_*}}{2}t}\n\\end{equation*}\nfor any $t\\geq t_0$.  We deduce that the right-hand side of~\\eqref{eq:varyingarc} is equal to $\\int_{M}f\\;\\text{d}\\vol+o(t)$ as $t$ tends to infinity. An elementary application of the Stone-Weierstrass' theorem (cf.~\\cite[Thm.~4.51]{Folland}) gives that smooth functions are dense in the space of continuous functions on the compact manifold $M$; it follows that the convergence\n\\begin{equation*}\n\t\\frac{1}{\\theta_2(t)-\\theta_1(t)}\\int_{\\theta_1(t)}^{\\theta_2(t)}f\\circ \\phi_t^{X}\\circ r_{s}(p)\\;\\text{d}s\\overset{t\\to\\infty}{\\longrightarrow}\\int_{M}f\\;\\text{d}\\vol\n\\end{equation*}\ncan be upgraded to hold for every $f\\in \\mathscr{C}(M)$, whereby the desired equidistribution is shown.\n\n\n\\subsubsection*{Equidistribution of circle arcs on the surface}\nWe conclude this subsection with a few comments concerning the statement of Theorem~\\ref{thm:expandingonsurface}, which is nothing but a specialization of Theorem~\\ref{thm:mainexpandingtranslates} to the case of observables defined on the underlying surface $S=\\Gamma\\bsl \\Hyp$, except for the lower regularity assumed on the test function $f$. First, we remark that $\\SO_2(\\R)$-invariance of the functions $D^{\\pm}_{4\\pi,\\mu}f$ and $\\mathcal{R}_{4\\pi}f$ follows at once from their definition (see~\\eqref{eq:defDthetamu} and~\\eqref{eq:remainder}) and the fact that $f$ is assumed to be $\\SO_2(\\R)$-invariant. We are only left to show that we might take $s>9\/2$, less restrictively in comparison to an arbitrary $f$\ndefined on $M$. The relevant observation here is that, for any $\\SO_2(\\R)$-invariant function $f\\in L^{2}(M)$, the components $f_{\\mu}$ appearing in the decomposition\\footnote{Recall from Section~\\ref{sec:unitaryrepresentations} that the Casimir operator $\\square$ acts as the Laplace-Beltrami operator $\\Delta_S$ on $\\SO_2(\\R)$-invariant functions.}\n\\begin{equation*}\n\tf=\\sum_{\\mu \\in \\text{Spec}(\\Delta_S)}f_{\\mu}\\;,\\quad f_{\\mu}\\in H_{\\mu}\n\\end{equation*} \nare invariant under $\\SO_2(\\R)$, that is, they satisfy $\\Theta f_{\\mu}=0$. The estimate in~\\eqref{eq:plustwo-three} thus only requires $s>9\/2=11\/2-1$, as the sum\n\\begin{equation*}\n\t\\sum_{\\mu \\in \\text{Spec}(\\Delta_S)}\\frac{1}{(1+\\mu)^{k}}\n\\end{equation*}\nconverges already for $k=2$, and not only for $k=3$ as it is the case in~\\eqref{eq:plustwo-three}.\n\n\\subsection{Equidistribution of arbitrary translates}\n\\label{sec:arbitrarytranslates}\n\nIn light of Theorem~\\ref{thm:mainexpandingtranslates}, Theorem~\\ref{thm:mainarbitrarytranslates} is a rather straightforward consequence of the classical Cartan decomposition for the semisimple Lie group $\\SL_2(\\R)$, for which the reader is referred to~\\cite[Chap.~VI]{Knapp}. We present the details of the argument in this subsection. \n\nLet $A=\\{\\exp{tX}:t\\in \\R\\}$ be the subgroup of $\\SL_2(\\R)$ consisting of diagonal matrices with positive entries (recall that $X$ is defined as in~\\eqref{eq:geodesicflow}). The product map\n\\begin{equation*}\n\t\\SO_2(\\R)\\times A\\times \\SO_2(\\R)\\to \\SL_2(\\R),\\;(k_1,a,k_2)\\mapsto k_1ak_2\n\\end{equation*} \nis surjective. For any $g\\in \\SL_2(\\R)$, choose a decomposition $g=k_1(g)a(g)k_2(g)$, where $a(g)$ is the diagonal matrix having as entries the singular values of the matrix $g$, in decreasing order. In particular, if $t(g)\\in \\R_{\\geq 0}$ is (uniquely) determined by the condition\n\\begin{equation}\n\t\\label{eq:Cartanprojection}\n\ta(g)=\n\t\\begin{pmatrix}\n\t\te^{t(g)\/2}&0\\\\\n\t\t0&e^{-t(g)\/2}\n\t\\end{pmatrix}\n\t,\n\\end{equation}\nthen it clearly holds that\n\\begin{equation}\n\t\\label{eq:singvaluenorm} \\norm{g}_{\\text{op}}=e^{t(g)\/2}\\;, \\quad \\text{ or equivalently} \\quad  t(g)=2\\log{\\norm{g}_{\\text{op}}}\\;.\n\\end{equation}\n\nFix now  a real number $s>11\/2$ and a function $f$ in the Sobolev space $W^{s}(M)$. Recall that, for any $p \\in M$, we indicate with $m_{\\SO_2(\\R)\\cdot p}$ the unique $\\SO_2(\\R)$-invariant measure supported on the compact orbit $\\SO_2(\\R)\\cdot p$; furthermore, $g_{*}\\SO_2(\\R)$ denotes the push-forward of the latter measure under the right translation map $R_{g}(\\Gamma g')=\\Gamma g'g$ on $M$. For any $p \\in M$ and $g\\in \\SL_2(\\R)$, we resort to the Cartan decomposition of $g$ and write\n\\begin{equation}\n\t\\label{eq:rotationinvariance}\n\t\\begin{split}\n\t\t\\int_{M}f\\;\\text{d}g_*m_{\\SO_2(\\R)\\cdot p}&=\\int_{M}f\\circ R_{g}\\;\\text{d}m_{\\SO_2(\\R)\\cdot p}=\\int_{M}f\\circ R_{k_2(g)}\\circ R_{a(g)}\\circ R_{k_1(g)}\\;\\text{d}m_{\\SO_2(\\R)\\cdot p}\\\\\n\t\t&=\\int_{M}f\\circ R_{k_2(g)}\\circ R_{a(g)}\\;\\text{d}m_{\\SO_2(\\R)\\cdot p}=\\frac{1}{4\\pi}\\int_0^{4\\pi}\\bigl(f\\circ R_{k_2(g)}\\bigr)\\circ \\phi^{X}_{t(g)}\\circ r_s(p)\\;\\text{d}s\\;,\n\t\\end{split}\n\\end{equation}\nusing the $R_{k_1(g)}$-invariance of $m_{\\SO_2(\\R)\\cdot p}$ and the fact that $R_{a(g)}=\\phi_{t(g)}^{X}$ in view of~\\eqref{eq:Cartanprojection}. \n\nWe may now make use of the asymptotic expansion provided by Theorem~\\ref{thm:mainexpandingtranslates} for the function $f\\circ R_{k_2(g)}$, which lies in the same Sobolev space $W^{s}(M)$ of $f$ since $R_{k_2(g)}$ is a smooth diffeomorphism of $M$. We thereby obtain, for a fixed base point $p \\in M$,\n\\begin{equation}\n\t\\label{eq:variabletranslates}\n\t\\begin{split}\n\t\t&\\frac{1}{4\\pi}\\int_0^{4\\pi}\\bigl(f\\circ R_{k_2(g)}\\bigr)\\circ \\phi^{X}_{t(g)}\\circ r_s(p)\\;\\text{d}s=\\int_{M}f\\circ R_{k_2(g)}\\;\\text{d}\\vol\\\\\n\t\t&+e^{-\\frac{t(g)}{2}}\\biggl( \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}t(g)\\biggr)} D^{+}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}t(g)\\biggr)} D^{-}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)\\biggr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}t(g)}D^{+}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)+e^{-\\frac{1-\\nu}{2}t(g)}D^{-}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)\\\\\n\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{t(g)}{2}}D^{+}_{4\\pi,1\/4}(f\\circ R_{k_2(g)})(p)+t(g)e^{-\\frac{t(g)}{2}}D^{-}_{4\\pi.1\/4}(f\\circ R_{k_2}(g))(p)\\bigr)+\\mathcal{R}_{4\\pi}(f\\circ R_{k_2(g)})(p,t(g))\n\t\\end{split}\n\\end{equation}\nfor any $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\text{op}}\\geq \\sqrt{e}$.\nDefine now, for any Casimir eigenvalue $\\mu \\in \\R_{>0}$, the functions $D^{\\pm}_{\\mu}\\colon M\\times \\SL_2(\\R)\\to \\C$ by \n\\begin{equation}\n\t\\label{eq:differentcoefficients}\n\tD^{\\pm}_{\\mu}f(p,g)=D^{\\pm}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)\\;,\\quad p \\in M,\\;g\\in \\SL_2(\\R),\n\\end{equation}\nand set also\n\\begin{equation*}\n\t\\mathcal{R}f(p,g)=\\mathcal{R}_{4\\pi}(f\\circ R_{k_2}(g))(p,t(g))\\;,\\quad p \\in M,\\;g\\in \\SL_2(\\R).\n\\end{equation*}\n\n\\smallskip\nThen, combining~\\eqref{eq:rotationinvariance} and~\\eqref{eq:variabletranslates} and recalling~\\eqref{eq:singvaluenorm} together with the fact that\\\\ $\\int_{M}f\\circ R_{k_2}\\;\\text{d}\\vol=\\int_{M}f\\;\\text{d}\\vol$, we deduce\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\int_{M}f\\;\\text{d}&g_{*}m_{\\SO_2(\\R)\\cdot p}=\\int_{M}f\\;\\text{d}\\vol\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\norm{g}^{-1}_{\\text{op}}\\bigl(\\cos{(\\Im{\\nu}\\log{\\norm{g}_{\\text{op}}})}D^{+}_{\\mu}f(p,g)+\\sin{(\\Im{\\nu}\\log{\\norm{g}_{\\text{op}}})}D^{-}_{\\mu}f(p,g)\\bigr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\norm{g}_{\\text{op}}^{-(1+\\nu)}D^{+}_{\\mu}f(p,g)+\\norm{g}_{\\text{op}}^{-(1-\\nu)}D^{-}_{\\mu}f(p,g)\\\\\n\t\t&+\\varepsilon_0\\bigl(\\norm{g}_{\\text{op}}^{-1}D^{+}_{1\/4}f(p,g)+2\\norm{g}_{\\text{op}}^{-1}\\log{\\norm{g}_{\\text{op}}}D^{-}_{1\/4}f(p,g)\\bigr)+\\mathcal{R}f(p,g)\n\t\\end{split}\n\\end{equation*}\nfor any $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\text{op}}\\geq \\sqrt{e}$,\nwhich is precisely the asymptotic expansion appearing in the statement of Theorem~\\ref{thm:mainarbitrarytranslates}.\n\nAs stated in Theorem~\\ref{thm:mainexpandingtranslates}, the functions $D^{\\pm}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})$ are continuous on $M$ for any fixed $\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}$ and $g\\in \\SL_2(\\R)$; equivalently, by~\\eqref{eq:differentcoefficients}, $D^{\\pm}_{\\mu}f(\\cdot,g)$ is continuous on $M$ for any fixed $g\\in \\SL_2(\\R)$.\n\nAlso, for any $p \\in M, \\;g\\in \\SL_2(\\R)$ and $\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}$, we have \n\\begin{equation*}\n\t|D^{\\pm}_{\\mu}f(p,g)|=|D^{\\pm}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})(p)|\\leq \\norm{D^{\\pm}_{4\\pi,\\mu}(f\\circ R_{k_2(g)})}_{\\infty}\\leq \\frac{C_{1,s-3}C_{\\mu}\\kappa(\\mu)}{4\\pi}\\norm{f\\circ R_{k_2(g)}}_{W^{s}}\\;,\n\\end{equation*} \nwhere the last inequality is given by~\\eqref{eq:boundDthetamu}. It remains to observe that compactness of $\\SO_2(\\R)$ implies that there exists a constant $C_{s,\\text{rot}}>0$ such that $\\norm{f\\circ R_{k}}_{W^{s}}\\leq C_{s,\\text{rot}}\\norm{f}_{W^s}$ for any $k\\in \\SO_2(\\R)$. The proof of this assertion runs along the same lines of the proof of Lemma~\\ref{lem:diffunderint}, with the appropriate modifications. Therefore, we get\n\\begin{equation*}\n\t\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}}\\sup_{p \\in M, \\;g\\in \\SL_2(\\R)}|D^{\\pm}_{\\mu}f(p,g)|\\leq \\frac{C_{1,s-3}C_{s,\\text{rot}}C'_{\\text{Spec}}}{4\\pi}\\norm{f}_{W^{s}}\\;,\n\\end{equation*} \nwhere $C'_{\\text{Spec}}$ is as in Theorem~\\ref{thm:mainexpandingtranslates}.\n\nTo conclude the proof of Theorem~\\ref{thm:mainarbitrarytranslates}, it is left to take care of the remainder term $\\mathcal{R}f$. We easily estimate, from~\\eqref{eq:globalremainderestimate},\n\\begin{equation*}\n\t\\begin{split}\n\t\t|\\mathcal{R}f(p,g)|&=|\\mathcal{R}_{4\\pi}(f\\circ R_{k_2}(g))(p,t(g))|\\leq \\frac{C_{\\text{Spec}}C_{1,s-3}}{4\\pi}\\norm{f\\circ R_{k_2(g)}}_{W^{s}}(t(g)+1)e^{-t(g)}\\\\\n\t\t&\\leq \\frac{C_{\\text{Spec}}C_{1,s-3}C_{s,\\text{rot}}}{4\\pi}\\norm{f}_{W^{s}}(2\\log{\\norm{g}_{\\text{op}}}+1)\\norm{g}_{\\text{op}}^{-2}\n\t\\end{split}\n\\end{equation*}\nfor any $p\\in M$ and $g\\in \\SL_2(\\R)$ with $\\norm{g}_{\\text{op}}\\geq \\sqrt{e}$.\n\nThis achieves the proof of Theorem~\\ref{thm:mainarbitrarytranslates}.\n\n\\section{Distributional limit theorems for deviations from the average}\n\\label{sec:CLT}\n\nThe purpose of this section is threefold, articulated in three subsections. First, we establish the quantitative distributional convergence claimed in Proposition~\\ref{prop:CLT}, from which the qualitative statements in Theorem~\\ref{thm:CLT} follow directly; secondly, we prove absence of a central limit theorem as phrased in Theorem~\\ref{thm:noCLT}, and finally we explore further ways of examining the statistical behaviour of averages along circle arcs. \n\n\\subsection{Quantitative distributional convergence}\n\\label{sec:spatialDLT}\nLet us fix the length parameter $\\theta\\in (0,4\\pi]$, and consider a real-valued function $f$ lying in the Sobolev space $W^{s}(M)$ for some real $s>11\/2$. We are interested in the statistical behaviour of the deviations from the mean \n\\begin{equation*}\n\td_f(T,p)=\\frac{1}{\\theta}\\int_{0}^{\\theta}f\\circ \\phi^{X}_T\\circ r_{s}(p)\\;\\text{d}s -\\int_{M}f\\;\\text{d}\\vol\n\\end{equation*}\nappropriately renormalized, as the time parameter $T$ tends to infinity and when the base point $p$ is sampled according to the uniform probability measure $\\vol$ on $M$.\nDefine\n\\begin{equation*}\n\t\\mu_f=\\inf\\{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}:D^{-}_{\\theta,\\mu}f\\text { does not vanish identically on }M  \\}\\;.\n\\end{equation*}\nAs in the hypotheses of Proposition~\\ref{prop:CLT}, we assume that $\\mu_f$ is finite, that is, the set of Casimir eigenvalues over which the previous infimum is taken is non-empty. Let $\\nu_f$ be the corresponding parameter, namely $\\nu_f\\in \\R_{\\geq 0}\\cup i \\R_{>0}$ satisfies $1-\\nu_f^{2}=4\\mu_f$.\n\nIn order to quantify the rate of distributional convergence of the random variables under consideration, we make use of the L\\'{e}vy-Prokhorov metric $d_{LP}$ on the set $\\mathscr{P}(\\R)$ of Borel probability measures on $\\R$. We recall that this is defined as \n\\begin{equation*}\n\td_{LP}(\\lambda,\\rho)=\\inf\\{\\varepsilon>0:\\lambda(Y)\\leq \\rho(Y_{\\varepsilon})+\\varepsilon \\text{ and }\\rho(Y)\\leq \\lambda(Y_{\\varepsilon})+\\varepsilon \\text{ for every Borel set } Y\\subset \\R \\}\n\\end{equation*}\nfor any $\\lambda,\\rho \\in \\mathscr{P}(\\R)$, where $Y_{\\varepsilon}$ denotes the open $\\varepsilon$-neighborhood of $Y$ with respect to the Euclidean metric on $\\R$. The distance $d_{LP}$ induces the topology of weak convergence of probability measures on $\\mathscr{P}(\\R)$, namely the coarsest topology for which the maps \n\\begin{equation*}\n\t\\mathscr{P}(\\R)\\ni \\lambda \\mapsto \\int_{\\R}\\varphi\\;\\text{d}\\lambda \\in \\R\\;, \\quad \\varphi\\colon \\R \\to \\R \\text{ continuous and bounded}\n\\end{equation*}\nare continuous.\n\nIn the forthcoming estimates we shall make use of the following trivial upper bound for the L\\'{e}vy-Prokhorov distance between the laws of two random variables defined on the same probability space and taking on nearby values almost surely.\n\n\n\\begin{lem}\n\t\\label{lem:LPnearbyvariables}\n\tLet $(\\Omega,\\cF,\\mathbf{P})$ be a probability space, $\\varepsilon>0$. Suppose $X,X'\\colon \\Omega\\to \\R$ are random variables satisfying\n\t$|X(\\omega)-X'(\\omega)|< \\varepsilon$ for $\\mathbf{P}$-almost every $\\omega\\in \\Omega$. If $\\lambda_{X}$ and $\\lambda_{Y}$ denote the laws of $X$ and $X'$, respectively, then $d_{LP}(\\lambda_{X},\\lambda_{X'})\\leq \\varepsilon$.\n\\end{lem}\n\\begin{proof}\n\tLet $A\\subset \\R$ be a Borel subset. The event $\\{X\\in A \\}$ is contained in the event $\\{X'\\in A_{\\varepsilon} \\}$, up to a $\\mathbf{P}$-negligible subset, by the assumption on the distance between $X$ and $X'$. Therefore, \n\t\\begin{equation*}\n\t\t\\lambda_{X}(A)=\\mathbf{P}(X\\in A)\\leq \\mathbf{P}(X'\\in A_{\\varepsilon})=\\lambda_{X'}(A_{\\varepsilon})<\\lambda_{X'}(A_{\\varepsilon})+\\varepsilon\\;;\n\t\\end{equation*}\n\ta similar inequality holds reversing the role of $X$ and $X'$, whence $d_{LP}(\\lambda_X,\\lambda_{X'})\\leq \\varepsilon$.\n\\end{proof}\n\nWe now proceed with the proof of Proposition~\\ref{prop:CLT} by distinguishing the three different cases $0<\\mu_f<1\/4\\;,\\mu_f=1\/4$ and $\\mu_f>1\/4$.\n\n\\medskip\nSuppose first $0<\\mu_f<1\/4$. We would then like to show that the random variables\n\\begin{equation*}\n\te^{\\frac{1-\\nu_f}{2}T}\\;d_f(T,p)\\;,\\quad p\\sim \\vol\n\\end{equation*}\nconverge in distribution, as $T$ tends to infinity, to the random variable $D^{-}_{\\theta,\\mu_f}f(p)$, $p\\sim \\vol$. Observe that, by virtue the asymptotic expansion in~\\eqref{eq:asymptoticgeneral} and the assumption on $\\mu_f$, we have\n\\begin{equation*}\n\t\\begin{split}\n\t\te^{\\frac{1-\\nu_f}{2}T}\\;d_f(T,p)-D^{-}_{\\theta,\\mu_f}f(p)&=\\sum\\limits_{\\mu\\in \\text{Spec}(\\square),\\;\\mu_f<\\mu<1\/4}e^{-\\frac{\\nu_f-\\nu}{2}T}D^{-}_{\\theta,\\mu}f(p)\\\\\n\t\t&+e^{-\\frac{\\nu_f}{2}T}\\biggl( \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{-}_{\\theta,\\mu}f(p)\\biggr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{\\nu_f+\\nu}{2}T}D^{+}_{\\theta,\\mu}f(p)\\\\\n\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{\\nu_f}{2}T}D^{+}_{\\theta,1\/4}f(p)+Te^{-\\frac{\\nu_f}{2}T}D^{-}_{\\theta.1\/4}f(p)\\bigr)+e^{\\frac{1-\\nu_f}{2}T}\\mathcal{R}_{\\theta}f(p,T)\\;,\n\t\\end{split}\n\\end{equation*}\nso that, because of the uniform bound in~\\eqref{eq:boundDthetamu}, we may estimate\n\\begin{equation*}\n\t\\biggl|e^{\\frac{1-\\nu_f}{2}T}\\;d_f(T,p)-D^{-}_{\\theta,\\mu_f}f(p)\\biggr|\\leq \\frac{C_{1,s-3}C'_{\\text{Spec}}}{\\theta} \\norm{f}_{W^s}Te^{-\\frac{\\nu_f-\\Re{\\nu_f^{\\text{next}}}}{2}T}\n\\end{equation*}\nfor any $p \\in M$ and $T\\geq 1$, where $\\nu_f^{\\text{next}}$ is the parameter corresponding to the smallest eigenvalue $\\mu_f^{\\text{next}}$ of the Casimir operator exceeding\\footnote{Observe that we may dispense with the additional factor $T$ in the upper bound whenever $\\nu_f^{\\text{next}}\\in \\R$.} $\\mu_f$. \n\nBy Lemma~\\ref{lem:LPnearbyvariables}, and recalling the definitions of $\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T)$ and $\\mathbf{P}_{\\theta,f}$ introduced in Section~\\ref{sec:introductionCLT}, we get\n\\begin{equation*}\n\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T),\\mathbf{P}_{\\theta,f})\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}'}{\\theta} \\norm{f}_{W^{s}}Te^{-\\eta_f T}\n\\end{equation*}\nfor $\\eta_f=\\frac{\\nu_f-\\Re{\\nu_f^{\\text{next}}}}{2}$.\n\n\\medskip\nSimilarly, if $\\mu_f=1\/4$, we readily obtain from~\\eqref{eq:asymptoticgeneral} that\n\\begin{equation*}\n\t\\begin{split}\n\t\tT^{-1}e^{\\frac{T}{2}}&\\;d_f(T,p)-D^{-}_{\\theta,1\/4}f(p)=T^{-1}\\biggl(D^{+}_{\\theta,1\/4}f(p)+\\sum_{\\mu \\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\biggl(\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)\\\\\n\t\t&+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)}D^{-}_{\\theta,\\mu}f(p)\\biggr)+\\sum_{\\mu \\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{\\nu}{2}T}D^{+}_{\\theta,\\mu}f(p)+e^{\\frac{T}{2}}\\mathcal{R}_{\\theta}f(p,t)\\biggr)\n\t\\end{split}\n\\end{equation*}\nfor any $p \\in M$ and $T\\geq 1$; recalling the definition of the constant $C_{\\text{pos}}$ in~\\eqref{eq:Cpos}, we deduce the bound\n\\begin{equation*}\n\t\\bigl|T^{-1}e^{\\frac{T}{2}}\\;d_f(T,p)-D^{-}_{\\theta,1\/4}f(p)\\bigr|\\leq \\frac{C_{1,s-3}C_{\\text{pos}}}{\\theta} \\norm{f}_{W^{s}}T^{-1}\\;,\n\\end{equation*}\nso that, again by Lemma~\\ref{lem:LPnearbyvariables},\n\\begin{equation*}\n\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T),\\mathbf{P}_{\\theta,f})\\leq \\frac{C_{1,s-3}C_{\\text{pos}}}{\\theta}\\norm{f}_{W^{s}}T^{-1}\n\\end{equation*} \nfor any $T\\geq 1$, as desired.\n\n\\medskip\nFinally, for $\\mu_f>1\/4$, we have from~\\eqref{eq:asymptoticgeneral} that\n\\begin{equation}\n\t\\label{eq:DLTabovequarter}\n\t\\begin{split}\n\t\t&e^{\\frac{T}{2}}d_{f}(T,p)-\\varepsilon_0D^{+}_{\\theta,1\/4}f(p)-\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)\n\t\t+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)}D^{-}_{\\theta,\\mu}f(p)\\\\\\\n\t\t&= \\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{\\nu}{2}T}D^{+}_{\\theta,\\mu}f(p)+e^{\\frac{T}{2}}\\mathcal{R}_{\\theta}f(p,T)\n\t\\end{split}\n\\end{equation} \nfor any $p \\in M$ and $T\\geq 1$, from which we deduce what follows. Let $\\mu_*=\\inf \\bigl(\\text{Spec}(\\square)\\cap \\R_{>0}\\bigr)$ be the spectral gap of $S=\\Gamma\\bsl \\Hyp$ and $\\nu_*$  the corresponding parameter:\n\\begin{itemize}\n\t\\item if $\\mu_{*}< 1\/4$, then~\\eqref{eq:DLTabovequarter} and~\\eqref{eq:boundDthetamu} give\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t&\\biggl|e^{\\frac{T}{2}}d_{f}(T,p)-\\varepsilon_0D^{+}_{\\theta,1\/4}f(p)-\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)\n\t\t\t+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)}D^{-}_{\\theta,\\mu}f(p)\\biggr|\\\\\n\t\t\t&\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}'}{\\theta}\\norm{f}_{W^{s}}e^{-\\frac{\\nu_*}{2}T}\\;,\n\t\t\\end{split}\n\t\\end{equation*}\n\twhence\n\t\\begin{equation*}\n\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}'}{\\theta}\\norm{f}_{W^{s}}e^{-\\frac{\\nu_*}{2}T}\n\t\\end{equation*}\n\tfor any $T\\geq 1$.\n\t\\item if $\\mu_*\\geq 1\/4$, then~\\eqref{eq:DLTabovequarter} and~\\eqref{eq:globalremainderestimate} give\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t&\\biggl|e^{\\frac{T}{2}}d_{f}(T,p)-\\varepsilon_0D^{+}_{\\theta,1\/4}f(p)-\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)} D^{+}_{\\theta,\\mu}f(p)\n\t\t\t+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}T\\biggr)}D^{-}_{\\theta,\\mu}f(p)\\biggr|\\\\\n\t\t\t&\\leq \\frac{C_{1,s-3}C_{\\text{Spec}}}{\\theta}\\norm{f}_{W^{s}}(T+1)e^{-\\frac{T}{2}}\\;;\n\t\t\\end{split}\n\t\\end{equation*}\n\twe deduce that\n\t\\begin{equation*}\n\t\td_{LP}(\\mathbf{P}_{\\theta,f}^{\\text{circ}}(T),\\mathbf{P}_{\\theta,f}(T))\\leq\\frac{C_{1,s-3}C_{\\text{Spec}}}{\\theta}\\norm{f}_{W^{s}}(T+1)e^{-\\frac{T}{2}}\n\t\\end{equation*} \n\tfor any $T\\geq 1$.\n\\end{itemize}\n\n\nThis completes the proof of Proposition~\\ref{prop:CLT}.\n\n\\subsection{Failure of a distributional limit theorem}\n\\label{sec:noCLT}\nWe now turn to the proof of Theorem~\\ref{thm:noCLT}. Once again, we consider a fixed length parameter $\\theta\\in (0,4\\pi]$ and a function $f\\in W^{s}(M)$ for some real $s>11\/2$. This time, we suppose that the coefficients \n$D^{\\pm}_{\\theta,\\mu}f$ vanish identically on $M$ for any Casimir eigenvalue $\\mu>0$. As a result, the asymptotic expansion provided in~\\eqref{eq:fullexpansion} reduces to\n\\begin{equation}\n\t\\label{eq:expansionCLT}\n\t\\begin{split}\n\t\t\\frac{1}{\\theta}\\int_{0}^{\\theta}f\\circ \\phi_T^{X}\\circ r_s(p)\\;\\text{d}s=&\\int_{M}f\\;\\text{d}\\vol +e^{-T}\\int_1^{T}\\sum_{n\\in I(0)}-G_{\\theta,n}f_{0,n}(p,\\xi)\\;\\text{d}\\xi\n\t\t\\\\\n\t\t&+\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,T)+\\mathcal{R}_{\\theta,\\text{d}}f(p,T)\\;,\n\t\\end{split}\n\\end{equation}\nfor any $p \\in M$ and $T\\geq 1$, where $\\mathcal{R}_{\\theta,\\text{d}}$ is defined in~\\eqref{eq:remainderneg}. \n\nThe estimates carried out in Section~\\ref{sec:proofexpandingtranslates} lead to the bound\n\\begin{equation}\n\t\\label{eq:remainderCLT}\n\t\\biggl|\\sum_{n\\in I(0)}\\mathcal{R}_{\\theta,0,n}f_{0,n}(p,T)+\\mathcal{R}_{\\theta,\\text{d}}f(p,T)\\biggr|\\leq \\frac{(8e\\pi+\\kappa_0)C_{1,s-3}C_{\\text{Spec},3}\\sup\\{1,C_{\\text{disc}}\\}}{\\theta}\\norm{f}_{W^{s}}e^{-T}\\;.\n\\end{equation}\nOn the other hand, by means of~\\eqref{eq:constantterm} we expand \n\\begin{equation*}\n\t\\begin{split}\n\t\t\\sum_{n\\in I(0)}-G_{\\theta,n}f_{0,n}(p,\\xi)=&\\frac{2}{\\theta(1-e^{-2\\xi})}\\sum_{n\\in I(0)}(Uf_{0,n}\\circ \\phi_{\\xi}^{X}(p)-Uf_{0,n}\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\\\\n\t\t&+\\frac{e^{-2\\xi}}{\\theta(1-e^{-2\\xi})^{2}}\\sum_{n\\in I(0)}2in(f_{0,n}\\circ \\phi_{\\xi}^{X}(p)-f_{0,n}\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\\\\n\t\t&-\\frac{e^{-\\xi}}{\\theta(1-e^{-2\\xi})^{2}}\\sum_{n\\in I(0)}\\int_0^{\\theta}f_{0,n}\\circ \\phi^{X}_{\\xi}\\circ r_{s}(p)\\;\\text{d}s\\\\\n\t\t&+\\frac{2e^{-\\xi}}{\\theta(1-e^{-2\\xi})}\\sum_{n\\in I(0)}\\int_0^{\\theta}Xf_{0,n}\\circ \\phi^{X}_{\\xi}\\circ r_s(p)\\;\\text{d}s\\;,\n\t\\end{split}\n\\end{equation*}\nfrom which\n\\begin{equation*}\n\t\\sum_{n\\in I(0)}-G_{\\theta,n}f_{0,n}(p,\\xi)=\\frac{2}{\\theta(1-e^{-2\\xi})}(Uf_0\\circ \\phi^{X}_{\\xi}(p)-Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))+\\mathcal{R}_{G}f(p,\\xi)\\;,\n\\end{equation*}\nwhere \n\\begin{equation}\n\t\\label{eq:remaindersecondCLT}\n\t\\begin{split}\n\t\t|\\mathcal{R}_{G}f(p,\\xi)|&\\leq \\frac{1}{(1-e^{-2})^{2}}\\biggl(\\norm{f}_{W^{s}}e^{-2\\xi}+\\norm{f}_{\\infty}e^{-\\xi}+2\\norm{f}_{\\cC^{1}}e^{-\\xi}(1-e^{-2\\xi})\\biggr)\\\\\n\t\t&\\leq \\frac{1+C_{0,s}+2C_{1,s}}{(1-e^{-2})^{2}}\\norm{f}_{W^{s}}e^{-\\xi}\\;,\n\t\\end{split}\n\\end{equation}\nusing the bound $1-e^{-2\\xi}\\geq 1-e^{-2}$ valid for any $\\xi\\geq 1$.\n\nLet now $(B_T)_{T>0}$ be a collection of positive real numbers such that $B_T\\to\\infty$ as $T\\to\\infty$. In light of~\\eqref{eq:expansionCLT},~\\eqref{eq:remainderCLT} and~\\eqref{eq:remaindersecondCLT}, and because of the assumption on $(B_T)_{T>0}$, the distributional limits of the random variables\n\\begin{equation*}\n\t\\frac{e^{T}\\bigl( \\frac{1}{\\theta}\\int_0^{\\theta}f\\circ\\phi^{X}_{T}\\circ r_{s}(p)\\;\\text{d}s-\\int_{M}f\\;\\text{d}\\vol\\bigr)}{B_T}\\;,\\quad p\\sim \\vol,\n\\end{equation*}\nas $T$ tends to infinity coincide with the distributional limits of the random variables\n\\begin{equation}\n\t\\label{eq:sameCLT} \\frac{\\frac{2}{\\theta}\\int_1^{T}\\frac{1}{1-e^{-2\\xi}}(Uf_0\\circ \\phi^{X}_{\\xi}(p)-Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\;\\text{d}\\xi}{B_T}\\;,\\quad p\\sim \\vol\\;.\n\\end{equation}\nWhen $\\theta=4\\pi$, we have $r_{4\\pi}(p)=p$, so that the integrand in the numerator of the above expression vanishes. Therefore, the distributional limit we are seeking after equals to zero almost surely, which proves Theorem~\\ref{thm:noCLT}.\n\n\\medskip\nAs to Remark~\\ref{rmk:geodesiccoboundary}, suppose $\\theta\\in (0,4\\pi]$ is arbitrary, and that $Uf_0$ is a coboundary for $(\\phi^{X}_t)_{t\\in \\R}$, namely there exists a measurable function $g\\colon M\\to \\C$ with\\footnote{More accurately, this is the notion of a measurable coboundary; by the celebrated work of Livsic on the cohomological equation for Anosov flows (cf.~\\cite{Livsic}), the condition is actually equivalent to the seemingly more restrictive one of $Uf_0$ being a continuous coboundary, namely of requiring the transfer function $g$ to be continuous.} finite norm\n\\begin{equation*}\n\t\\norm{g}_{L^{\\infty}(M,\\vol)}=\\inf\\{\\lambda \\in \\R_{>0}:|g(p)|\\leq \\lambda \\text{ for $\\vol$-almost every }p \\in M  \\}\n\\end{equation*}\n\nsuch that, for all $T>0$, \n\\begin{equation*}\n\t\\int_0^{T}Uf_0\\circ \\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi=g\\circ \\phi^{X}_T(p)-g(p) \\quad \\text{for $\\vol$-almost every }p \\in M\\;.\n\\end{equation*}\n\nIt follows trivially that, for every $T>0$,\n\\begin{equation*}\n\t\\biggl|\\int_1^{T}\\frac{1}{1-e^{-2\\xi}}(Uf_0\\circ \\phi^{X}_{\\xi}(p)-Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\;\\text{d}\\xi\\biggr|\\leq \\frac{4}{1-e^{-2}}\\norm{g}_{L^{\\infty}(M,\\vol)} \n\\end{equation*}\nfor $\\vol$-almost every $p \\in M$.\nAs a result, the distributional limit as $T\\to\\infty$ of the random variables in~\\eqref{eq:sameCLT} vanishes almost surely, since $B_T\\to\\infty$.\n\n\\begin{rmk}\n\tSlightly more generally, when $Uf_0$ is cohomologous to a constant function, namely it differs from a constant function by a coboundary, any distributional limit of the random variables in~\\eqref{eq:sameCLT} is almost surely constant.\n\t\n\tAssume now $Uf_0$ is not cohomologous to a constant function (and $\\theta\\neq 4\\pi$). The classical central limit theorem for geodesic ergodic integrals (see~\\cite{Sinai} for the constant curvature case, and~\\cite{Ratner} for variable negative curvature) gives that both\n\t\\begin{equation*}\n\t\t\\frac{\\int_1^{T}Uf_0\\circ \\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi-\\int_{M}Uf_0\\;\\text{d}\\vol}{\\sqrt{T}}\\;,\\quad p\\sim \\vol\n\t\\end{equation*}\n\tand \n\t\\begin{equation*}\n\t\t\\frac{\\int_1^{T}Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)\\;\\text{d}\\xi-\\int_{M}Uf_0\\;\\text{d}\\vol}{\\sqrt{T}}\\;,\\quad p\\sim \\vol\n\t\\end{equation*}\n\tconverge in distribution to a non-trivial centered Gaussian random variable as $T$ tends to infinity. \\textit{A priori}, the combination of these two distributional convergences doesn't provide any information on the distributional limits of the difference, which is what appears in~\\eqref{eq:sameCLT} up to the constant factor $2\/\\theta$; it would be desirable to reach a full understanding of this limiting distributional behaviour by carefully inspecting the dependence properties of the random variables $\\int_1^{T}Uf_0\\circ \\phi^{X}_\\xi(p)\\;\\text{d}\\xi$ and $\\int_1^{T}Uf_0\\circ \\phi^{X}_\\xi\\circ r_{\\theta}(p)\\;\\text{d}\\xi$ as $p$ is sampled according to the volume measure on $M$. \n\\end{rmk}\n\n\\subsection{Some reflections on temporal distributional limit theorems}\n\\label{sec:temporalDLT}\nAn upshot of the two foregoing subsections is the following consideration: examining the statistical behaviour, for large times $T$, of the (appropriately renormalized) averages \n\\begin{equation*}\n\t\\frac{1}{\\theta}\\int_{0}^{\\theta}f\\circ \\phi^{X}_{T}\\circ r_s(p)\\;\\text{d}s\n\\end{equation*}\nby randomly sampling the base point $p$ according to the uniform measure on $M$ leads to meaningful asymptotic results if and only if\\footnote{Possibly with the exception of the case examined at the end of Section~\\ref{sec:noCLT}.} at least one of the coefficients $D^{\\pm}_{\\theta,\\mu}f$ does not vanish identically on $M$. Irrespective of whether this is the case or not, it is natural to look for different sources of randomness, which might capture oscillatory behaviours more accurately. In accordance with the perspective of temporal distributional limit theorems, pioneered by Dolgopyat and Sarig~\\cite{Dolgopyat-Sarig} in the context of ergodic sums and integrals, we enquire about the existence of non-trivial distributional limits for the random variables\n\\begin{equation*}\n\t\\frac{e^{t}\\bigl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_t\\circ r_s(p)\\;\\text{d}s\\bigr)-A_T}{B_T}\\;,\n\\end{equation*}\nwhere $p$ is a fixed base point in $M$, $(A_T)_{T>0}$ and $(B_T)_{T>0}$ are collections of real numbers, possibly depending on $p$, with $B_T>0$ and $B_T\\to\\infty$ as $T\\to\\infty$, and the time $t$ is chosen uniformly at random in the interval $[0,T]$. \n\n\\begin{rmk}\n\tIt is informative to compare this to the quest for temporal limit theorems for ergodic integrals along the orbits of a flow: see, in particular,~\\cite[Def.~1.3]{Dolgopyat-Sarig}. Observe notably that the rescaling of the circle-arc average $\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi_t^{X}\\circ r_s(p)\\;\\text{d}s$ by a factor of $e^{t}$ (the latter being asymptotically of the same order of the length of the expanding circle arc along which the average is taken) parallels the renormalization of ergodic averages by the linear factor $t$.\n\\end{rmk}\n\nLet us denote by $\\mathcal{U}_{[0,T]}$ the uniform probability measure on the compact interval $[0,T]$, for any $T>0$. If there is a non-identically vanishing coefficient $D^{\\pm}_{\\theta,\\mu}f$ for some Casimir eigenvalue $\\mu>0$, then a rather straightforward adaptation of the proof of~\\cite[Cor.~5.7]{Dolgopyat-Sarig} shows that, for $\\vol$-almost every $p \\in M$, any limiting distribution of \n\\begin{equation}\n\t\\label{eq:temporalDLT}\n\t\\frac{e^{t}\\bigl(\\frac{1}{\\theta}\\int_0^{\\theta}f\\circ \\phi^{X}_t\\circ r_s(p)\\;\\text{d}s\\bigr)-A_T}{B_T}\\;,\\quad t\\sim \\mathcal{U}_{[0,T]}\n\\end{equation}\nis necessarily constant almost surely, no matter the choice of the constants $A_T$ and $B_T$.\n\nSuppose now that the coefficients $D^{\\pm}_{\\theta,\\mu}f$ vanish identically on $M$ for any  positive Casimir eigenvalue $\\mu$. The deduction in Section~\\ref{sec:noCLT} applies almost verbatim, showing  that the distributional limits of the random variables in~\\eqref{eq:temporalDLT} are the same as the limits of \n\\begin{equation}\n\t\\label{eq:sametemporal}\n\t\\frac{\\frac{2}{\\theta}\\int_1^{t}\\frac{1}{1-e^{-2\\xi}}(Uf_0\\circ \\phi^{X}_{\\xi}(p)-Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p))\\;\\text{d}\\xi-A_T}{B_T}\\;,\\quad t\\sim \\mathcal{U}_{[0,T]}\n\\end{equation}\nas $T$ tends to infinity. In the first place, this allows tu rule out the existence of any non-trivial (namely not almost surely constant) distributional limit whenever one of the following conditions is met:\n\\begin{itemize}\n\t\\item[(a)] $\\theta=4\\pi$;\n\t\\item[(b)] $Uf_0$ is cohomologous to a constant function for the geodesic flow.\n\\end{itemize} \nOn the other hand, when $Uf_0$ is not cohomologous to a constant function, then the geodesic ergodic integrals $\\int_1^{t}Uf_0\\circ \\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi$ are well-approximated by Brownian trajectories. More precisely, the Almost Sure Invariance Principle (see~\\cite{Strassen,Strassen-two},~\\cite[Chap.~1]{Philipp-Stout} and~\\cite{Denker-Phillip}) for geodesic ergodic integrals asserts that there exist an auxiliary probability space $(\\Omega,\\cF,\\mathbf{P})$ and two continuous-time stochastic processes $(X_t)_{t\\geq 0}$ and $(B_t)_{t\\geq 0}$ defined on $(\\Omega,\\cF,\\mathbf{P})$ such that the following hold:\n\\begin{itemize}\n\t\\item the law of the process $(X_{t})_{t\\geq 0}$ under the probability measure $\\mathbf{P}$ coincides with the law of the process $\\bigl(\\int_0^{t}Uf_0\\circ \\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi\\bigr)_{t\\geq 0}$ when $p$ is sampled according to the probability measure $\\vol$;\n\t\\item the process $(B_{t})_{t\\geq 0}$ is a standard one-dimensional Brownian motion (cf.~\\cite[Chap.~2]{LeGall});\n\t\\item there exists $\\sigma \\in \\R^{\\times}$ such that, for $\\mathbf{P}$-almost every $\\omega \\in \\Omega$,\n\t\\begin{equation}\n\t\t\\label{eq:Brownian}\n\t\t|X_t(\\omega)-B_{\\sigma^{2}t}(\\omega)|=o(\\sqrt{t}) \\quad \\text{as $t\\to\\infty$}\\;.\n\t\\end{equation}\n\\end{itemize}\nAs typical Brownian trajectories are of size $\\sim\\sqrt{t}$ at time $t$, the approximation in~\\eqref{eq:Brownian} enables to transfer classical results about the statistical behaviour of Brownian paths to analogous properties for geodesic ergodic integrals. In particular, there is no distributional limit\\footnote{Actually, when $Uf_0$ has zero average over $M$, $A_T=0$ and $B_T=\\sqrt{T}$,  any random variable may appear as distributional limit along an appropriate subsequence $(T_n)_{n\\in \\N}$ of times: see~\\cite[Thm.~3.2]{Dolgopyat-Sarig}.} for \n\\begin{equation}\n\t\\label{eq:temporalfirst}\n\t\\frac{\\int_0^{t}Uf_0\\circ\\phi^{X}_{\\xi}(p)\\;\\text{d}\\xi-A_T}{B_T}\\;,\\quad t\\sim\\mathcal{U}_{[0,T]}\n\\end{equation} \nas $T$ tends to infinity (cf.~\\cite[Sec.~3.1]{Dolgopyat-Sarig}). \nSince the process $\\bigl(\\int_0^{t}Uf_0\\circ \\phi^{X}_{\\xi}\\circ r_{\\theta}(p)\\;\\text{d}\\xi\\bigr)_{t\\geq 0}$ has the same law, for $p\\sim \\vol$, as $(X_t(\\omega))_{t\\geq 0}$ for $\\omega\\sim \\mathbf{P}$, the same applies to the random variables \n\\begin{equation}\n\t\\label{eq:temporalsecond}\n\t\\frac{\\int_0^{t}Uf_0\\circ\\phi^{X}_{\\xi}\\circ r_{\\theta}(p)\\;\\text{d}\\xi-A_T}{B_T}\\;,\\quad t\\sim\\mathcal{U}_{[0,T]}\\;.\n\\end{equation}\nAs already argued in Section~\\ref{sec:noCLT} in the situation where the point $p$ is selected randomly and the time $T$ is fixed, here again the absence of distributional limits for each of the summands does not rule out, in principle, the possibility of non-trivial limits for the difference, hence for~\\eqref{eq:sametemporal}. Once more, a painstaking analysis of the dependence features of the two processes in~\\eqref{eq:temporalfirst} and~\\eqref{eq:temporalsecond} might clarify the seemingly elusive pathwise behaviour of their difference.\n\n\\section{The hyperbolic lattice point counting problem}\n\\label{sec:latticepoint}\n\nThis final section is consecrated to the applications of our equidistribution results to lattice-point counting problems in the hyperbolic plane; specifically, we shall first prove the precise asymptotics for the averaged counting function stated in Proposition~\\ref{prop:averagedcounting} and subsequently deduce Theorem~\\ref{thm:countingproblem} on the error estimate for the pointwise counting.  \n\nLet $\\Gamma$ be a cocompact lattice in $\\SL_2(\\R)$, and denote by $d_{\\Hyp}$ the hyperbolic distance function on the hyperbolic upper-half plane $\\Hyp$ (cf.~Section~\\ref{sec:hyperbolic}). For each real number $R>0$, let $B_R$ be the closed $d_{\\Hyp}$-ball of radius $R$ centered at the point $i\\in \\Hyp$, and define $N(R)=|\\Gamma\\cdot i \\;\\cap B_R|$, the cardinality of intersection of the $\\Gamma$-orbit of $i$ with $B_R$.\n\nRecall also from Section~\\ref{sec:hyperbolic} that $\\SL_2(\\R)$ acts on $\\Hyp$ by M\\\"{o}bius transformations.\nIn what follows, we identity the quotient manifold $\\SL_2(\\R)\/\\SO_2(\\R)$ with $\\Hyp$ whenever convenient, by means of the diffeomorphism $g\\SO_2(\\R)\\mapsto g\\cdot i,\\;g\\in \\SL_2(\\R)$. The hyperbolic area measure $m_{\\Hyp}$ (namely the volume measure arising from the hyperbolic structure on $\\Hyp$) is the Radon measure on $\\Hyp$ with density $\\text{d}m_{\\Hyp}(x,y)=y^{-2}\\text{d}x\\text{d}y$ with respect to the induced Lebesgue measure on $\\Hyp\\subset \\C$. \n\n\\begin{term}\n\tIn order not to overburden notation in the sequel, we shall denote $\\SL_2(\\R)$ by $G$ and $\\SO_2(\\R)$ by $K$.\n\\end{term}\n\n\\subsection{Asymptotics for the averaged counting function}\n\\label{sec:averagecounting}\n\nFor any subset $A\\subset G\/K$, we denote by $\\mathds{1}_{A}$ the indicator function of the set $A$. Define a function $F_R\\colon G\/\\Gamma\\to \\R_{\\geq 0}$\n\\begin{equation}\n\t\\label{eq:defaveragedcounting}\n\tF_R(g\\Gamma )=\\frac{|g\\Gamma \\cdot i \\cap B_R|}{m_{\\Hyp}(B_R)}=\\frac{1}{m_{\\Hyp}(B_R)}\\sum_{\\gamma \\Gamma\\cap K\\in \\Gamma\/\\Gamma\\cap K}\\mathds{1}_{B_R}(g\\gamma K)\\;, \\quad g\\in G;\n\\end{equation}\nobserve that the function $F_R$ is the subject of the averaged counting result in Proposition~\\ref{prop:averagedcounting}, which we now set out to prove. \n\n\\begin{rmk}\n\t\\label{rmk:leftrightcosets}\n\tWe choose to deal with spaces of left cosets in the sequel; in particular, we replace the homogeneous spaces $M=\\Gamma\\bsl G$ we have been considering so far with $G\/\\Gamma$, identifying them via the diffeomorphism $\\Gamma g\\mapsto g^{-1}\\Gamma$.\n\\end{rmk}\n\nWe follow the classical argument of Eskin and McMullen \\cite{Eskin-McMullen}, which relies on the well-known folding-unfolding formula for invariant measures on homogeneous spaces. For the sake of completeness, we recall it in the setting of the group $G=\\SL_2(\\R)$, referring the reader to~\\cite[Sec.~2.6]{Folland} or to~\\cite[Chap.~1]{Raghunathan} for the general statements and their proofs. \n\n\\begin{prop}\n\t\\label{prop:foldingunfolding}\n\t\\begin{enumerate}\n\t\t\\item Let $H<G$ be a unimodular closed subgroup. Then there exists a non-zero $G$-invariant positive Radon measure on the quotient space $G\/H $, which is uniquely determined up to positive real scalars. Moreover, if $\\mu_{G}$ and $\\mu_H$ are Haar measures\\footnote{The group $G$ is perfect, hence unimodular; thus $\\mu_{G}$ is also a right Haar measure.} on $G$ and $H$, respectively, there exists a unique normalization $\\mu_{G\/H}$ of the $G$-invariant measure on $G(H)$ such that, for any continuous compactly supported function $\\varphi\\colon G\\to \\C$, the following folding-unfolding formula holds:\n\t\t\\begin{equation}\n\t\t\t\\label{eq:foldingunfolding}\n\t\t\t\\int_{G}\\varphi\\;\\emph{d}\\mu_{G}=\\int_{G\/H}\\int_{H}\\varphi(gh)\\;\\emph{d}\\mu_{H}(h)\\;\\emph{d}\\mu_{G\/H}(gH)\\;.\n\t\t\\end{equation}\n\t\t\\item Let $H<L$ be closed subgroups of $G$, and suppose $G\/L$ admits a non-zero finite $G$-invariant measure $\\mu_{G\/L}$ and $L\/H$ admits a non-zero finite $L$-invariant measure $\\mu_{L\/H}$. Then $G\/H$ admits a non-zero finite $G$-invariant measure $\\mu_{G\/H}$. Moreover, if $\\mu_{G\/L},\\;\\mu_{L\/H}$ and $\\mu_{G\/H}$ are compatibly normalized, then for any continuous compactly supported function $\\varphi\\colon G\/H\\to \\C$, it holds\n\t\t\\begin{equation}\n\t\t\t\\label{eq:foldingunfoldingchain}\n\t\t\t\\int_{G\/H}\\varphi\\;\\emph{d}\\mu_{G\/H}=\\int_{G\/L}\\int_{L\/H}\\varphi(glH)\\;\\emph{d}\\mu_{L\/H}(lH)\\;\\emph{d}\\mu_{G\/L}(gL)\\;.\n\t\t\\end{equation} \n\t\\end{enumerate}\n\\end{prop}\nBy means of standard approximation arguments in measure theory, formula~\\eqref{eq:foldingunfolding} (resp.~ formula~\\eqref{eq:foldingunfoldingchain}) holds for any Borel-measurable function $\\varphi\\colon G\\to \\C$ (resp.~$\\varphi\\colon G\/H\\to \\C$) which either takes positive real values or is integrable with respect to $\\mu_{G}$ (resp.~$\\mu_{G\/H}$). \n\n\\smallskip\nLet now $m_{K}$ be the unique probability Haar measure on the compact group $K$, and normalize the Haar measure $m_{G}$ on $G$ so that, under the identification of $G\/K$ with $\\Hyp$, the resulting $G$-invariant measure on $G\/K$ (cf.~Proposition~\\ref{prop:foldingunfolding}) corresponds to the hyperbolic area measure $m_{\\Hyp}$. Let $m_{G\/\\Gamma}$ be the unique $G$-invariant finite Borel measure on $G$ corresponding, according to Proposition~\\ref{prop:foldingunfolding}, to the given choice of Haar measure on $G$ and to the counting measure on the discrete group $\\Gamma$. Similarly, endowing the finite discrete group $\\Gamma\\cap K$ with the counting measure, we indicate with $m_{G\/\\Gamma\\cap K}$, $m_{K\/\\Gamma\\cap K}$ and $m_{\\Gamma\/\\Gamma\\cap K}$ the induced measures on the respective homogeneous spaces. Recall also that with $m_{K\\cdot \\Gamma}$ we indicate the unique $K$-invariant probability measure supported on the compact $K$-orbit of the identity coset $\\Gamma$ inside $G\/\\Gamma$ (cf.~Theorem~\\ref{thm:mainarbitrarytranslates}).\n\nThe volumes of the homogeneous spaces $G\/\\Gamma$ and $K\/\\Gamma\\cap K$ with respect to the measures $m_{G\/\\Gamma}$ and $m_{K\/\\Gamma\\cap K}$ are indicated with $\\text{covol}_{K}(\\Gamma\\cap K)$ and $\\text{covol}_{G}(\\Gamma)$, respectively.\n\n\\medskip\nFix now a real parameter $s>11\/2$ and a test function $\\psi\\in W^{s}(G\/\\Gamma)$. We expand, for any $R>0$,\n\n\\begin{equation}\n\t\\label{eq:foldunfold}\n\t\\begin{split}\n\t\t&\\int_{M}\\psi F_R\\;\\text{d}m_{G\/\\Gamma}=\\int_{M}\\psi(g\\Gamma )\\biggl(\\frac{1}{m_{\\Hyp}(B_R)}\\sum_{\\gamma \\Gamma\\cap K\\in \\Gamma\/\\Gamma\\cap K}\\mathds{1}_{B_R}(g\\gamma K)\\biggr) \\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)\\\\\n\t\t&=\\frac{1}{m_{\\Hyp}(B_R)}\\int_{M}\\int_{\\Gamma\/\\Gamma\\cap K}\\psi(g\\Gamma )\\mathds{1}_{B_R}(g\\gamma K)\\;\\text{d}m_{\\Gamma\/\\Gamma\\cap K}(\\gamma\\Gamma\\cap K) \\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)\\\\\n\t\t&=\\frac{1}{m_{\\Hyp}(B_R)}\\int_{G\/\\Gamma\\cap K} \\psi(g\\Gamma)\\mathds{1}_{B_R}(gK) \\text{d}m_{G\/\\Gamma\\cap K}(g\\Gamma\\cap K)\\\\\n\t\t&=\\frac{1}{m_{\\Hyp}(B_R)}\\int_{G\/K}\\int_{K\/\\Gamma\\cap K}\\psi(gk\\Gamma)\\mathds{1}_{B_R}(gK)\\;\\text{d}m_{K\/\\Gamma\\cap K}(k\\Gamma\\cap K)\\;\\text{d}m_{G\/K}(gK)\\\\\n\t\t&=\\frac{1}{m_{\\Hyp}(B_R)}\\int_{B_R}\\int_{K\/\\Gamma\\cap K}\\psi\\;\\text{d}g_{*}m_{K\/\\Gamma\\cap K}\\;\\text{d}m_{\\Hyp}(gK)\\\\\n\t\t&=\\frac{\\text{covol}_{ K}(\\Gamma\\cap K )}{m_{\\Hyp}(B_R)}\\int_{B_R}\\int_{G\/\\Gamma}\\psi\\;\\text{d}g_{*}m_{K\\cdot \\Gamma}\\;\\text{d}m_{\\Hyp}(gK)\\;.\n\t\\end{split}\n\\end{equation}\nIn the previous chain of equalities, we applied in successive order:\n\\begin{itemize}\n\t\\item[(1)] the definition~\\eqref{eq:defaveragedcounting} of the function $F_R$;\n\t\\item[(2)] the fact that the invariant measure on the discrete space $\\Gamma\/\\Gamma\\cap K$ given by Proposition~\\ref{prop:foldingunfolding} is the counting measure;\n\t\\item[(3)] formula~\\eqref{eq:foldingunfoldingchain} to the tower of subgroups $\\Gamma\\cap K<\\Gamma<G$;\n\t\\item[(4)] formula~\\eqref{eq:foldingunfoldingchain} to the tower of subgroups $\\Gamma\\cap K<K<G$;\n\t\\item[(5)] the identification of $\\Hyp$ with $G\/K$ and of $m_{\\Hyp}$ with $m_{G\/K}$;\n\t\\item[(6)] the relationship $m_{K\/\\Gamma\\cap K}=\\text{covol}_{K}(\\Gamma\\cap K)m_{K\\cdot \\Gamma}$, derived from the fact that $m_{K\\cdot \\Gamma}$ is a probability measure, the definition of $\\text{covol}_{K}(\\Gamma\\cap K)$ and uniqueness up to scalars of the $K$-invariant measure on $K\/\\Gamma \\cap K\\simeq K\\cdot \\Gamma$.  \n\\end{itemize}\n\nWe may now replace the inner integral in the last expression of~\\eqref{eq:foldunfold} with the asymptotic expansion provided by Theorem~\\ref{thm:mainarbitrarytranslates} (with the caveat of Remark~\\ref{rmk:leftrightcosets}), thereby obtaining\n\\begin{equation}\n\t\\label{eq:asymptoticaverage}\n\t\\begin{split}\n\t\t\\int_{G\/\\Gamma}\\psi F_R&\\;\\text{d}m_{G\/\\Gamma}=\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}\\int_{G\/\\Gamma}\\psi\\;\\text{d}m_{G\/\\Gamma} +\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{m_{\\Hyp}(B_R)}\\\\\n\t\t&\\int_{B_R}\\biggl(\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\norm{g}^{-1}_{\\text{op}}\\bigl(\\cos{(\\Im{\\nu}\\log{\\norm{g}_{\\text{op}}})}D^{+}_{\\mu}\\psi(\\Gamma,g)+\\sin{(\\Im{\\nu}\\log{\\norm{g}_{\\text{op}}})}D^{-}_{\\mu}\\psi(\\Gamma,g)\\bigr)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\norm{g}_{\\text{op}}^{-(1+\\nu)}D^{+}_{\\mu}\\psi(\\Gamma,g)+\\norm{g}_{\\text{op}}^{-(1-\\nu)}D^{-}_{\\mu}\\psi(\\Gamma,g)\\\\\n\t\t&+\\varepsilon_0\\bigl(\\norm{g}_{\\text{op}}^{-1}D^{+}_{1\/4}\\psi(\\Gamma,g)+2\\norm{g}_{\\text{op}}^{-1}\\log{\\norm{g}_{\\text{op}}}D^{-}_{1\/4}\\psi(\\Gamma,g)\\bigr)+\\mathcal{R}\\psi(\\Gamma,g)\\biggr) \\text{d}m_{\\Hyp}(gK)\\;.\n\t\\end{split}\n\\end{equation}\n\nWe shall need the following analogue of the classical integration formula on spheres in Euclidean spaces: for any $r>0$, let $S_r=\\partial{B_r}=\\{z\\in \\Hyp:d_{\\Hyp}(z,i)=r \\}$ and $\\sigma_r$ the induced hyperbolic length measure on the circle $S_r$.\n\n\\begin{prop}\n\t\\label{prop:sphereintegration}\n\tLet $f\\colon \\Hyp\\to \\C$ be integrable with respect to $m_{\\Hyp}$. Then\n\t\\begin{equation*}\n\t\t\\int_{\\Hyp}f\\;\\emph{d}m_{\\Hyp}=\\int_{0}^{\\infty}\\int_{S_r}f(z)\\;\\emph{d}\\sigma_{r}(z)\\;\\emph{d}r\n\t\\end{equation*}\n\\end{prop} \n\nThe proof does not differ from the Euclidean case, for which we refer to~\\cite[Thm.~2.49]{Folland}.\n\n\\medskip\nDefine now, for any $\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}$ and $\\psi$ as above,\n\\begin{equation*}\n\t\\alpha^{\\pm}_{\\psi,\\mu}(r)=\\int_{S_r}D^{\\pm}_{\\mu}\\psi(\\Gamma,z)\\;\\text{d}\\sigma_r(z)\\;,\\quad r>0.\n\\end{equation*}\nFrom~\\eqref{eq:asymptoticaverage} we get, thanks to Proposition~\\ref{prop:sphereintegration}, \n\\begin{equation}\n\t\\label{eq:applyingsphereintegration}\n\t\\begin{split}\n\t\t&\\int_{G\/\\Gamma}\\psi F_R\\;\\text{d}m_{G\/\\Gamma}=\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}\\int_{G\/\\Gamma}\\psi\\;\\text{d}m_{G\/\\Gamma}+\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{m_{\\Hyp}(B_R)}\\\\\n\t\t&\\biggl(\\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\int_0^{R}e^{-\\frac{r}{2}}\\biggl(\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}r\\biggr)}\\alpha_{\\psi,\\mu}^{+}(r)+\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}r\\biggr)}\\alpha^{-}_{\\psi,\\mu}(r)\\biggr)\\text{d}r\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}\\int_0^{R}e^{-\\frac{1+\\nu}{2}r}\\alpha^{+}_{\\psi,\\mu}(r)+e^{-\\frac{1-\\nu}{2}r}\\alpha^{-}_{\\psi,\\mu}(r)\\;\\text{d}r\\\\\n\t\t&+\\varepsilon_0\\biggl(\\int_0^{R}e^{-\\frac{r}{2}}\\alpha^{+}_{\\psi,1\/4}(r)+re^{-\\frac{r}{2}}\\alpha^{-}_{\\psi,1\/4}(r)\\;\\text{d}r\\biggr)+\\int_0^{R}\\int_{S_r}\\mathcal{R}\\psi(\\Gamma,z)\\;\\text{d}\\sigma_r(z)\\;\\text{d}r\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\nLet $\\psi_{\\mu}$ be the orthogonal projection of $\\psi$ onto the closed subspace $W^{s}(H_{\\mu})$. By means of~\\eqref{eq:boundDthetamu}, we estimate \n\\begin{equation}\n\t\\label{eq:integralbounds}\n\t\\sum_{\\mu \\in \\text{Spec}(\\square)\\cap \\R_{>0}}|\\alpha_{\\psi,\\mu}^{\\pm}(r)|\\leq \\sum_{\\mu\\in \\text{Spec}(\\square)\\cap \\R_{>0}}\\norm{D^{\\pm}_{\\mu}\\psi(\\Gamma,\\cdot )}_{\\infty}\\int_{S_r}\\text{d}\\sigma_{r}(z)\\leq 2\\pi \\frac{C_{1,s-3}C_{\\text{Spec}}'}{4\\pi}\\norm{\\psi}_{W^{s}}\\sinh{r}\n\\end{equation}\nfor any $r>0$,\nas the hyperbolic length  of $S_r$ equals\\footnote{This is an elementary verification in hyperbolic geometry, for instance approximating circles with regular $n$-gons; their hyperbolic perimeter can be easily computed by means of explicit formulas for the hyperbolic distance (cf.~\\cite[Thm.~1.2.6]{Katok}) and of the hyperbolic cosine law (cf.~\\cite[Thm.~1.5.2]{Katok}). \n\t\n\tSimilarly, the hyperbolic area of a ball is easily computed by approximation via the Gauss-Bonnet formula for the area of hyperbolic triangles (cf.~\\cite[Thm.~1.4.2]{Katok}).} $2\\pi\\sinh{r}$. Recalling that $m_{\\Hyp}(B_R)=2\\pi(\\cosh{R}-1)$ for any $R>0$, we deduce from~\\eqref{eq:applyingsphereintegration} that\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\int_{G\/\\Gamma}\\psi F_R\\;\\text{d}m_{G\/\\Gamma}=&\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}\\int_{G\/\\Gamma}\\psi\\;\\text{d}m_{G\/\\Gamma}\\\\\n\t\t&+\\text{covol}_{K}(\\Gamma\\cap K)\n\t\t\\biggl(e^{-\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^{+}_{\\psi,\\mu}(R)+\\beta^{-}_{\\psi,\\mu}(R)\\\\\n\t\t&\n\t\t+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}R}\\beta^{+}_{\\psi,\\mu}(R)+e^{-\\frac{1-\\nu}{2}R}\\beta^{-}_{\\psi,\\mu}(R)\\\\\n\t\t&+\\varepsilon_0\\bigl(e^{-\\frac{R}{2}}\\beta^{+}_{\\psi,1\/4}(R)+Re^{-\\frac{R}{2}}\\beta^{-}_{\\psi,1\/4}(R)\\bigr)+\\gamma_{\\psi}(R)\\biggr)\n\t\\end{split}\n\\end{equation*}\nfor any $R\\geq 1$, where we have set\n\\begin{align*}\n\t&\\beta^+_{\\psi,\\mu}(R)=\\frac{e^{-\\frac{R}{2}}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{r}{2}}\\cos{\\biggl(\\frac{\\Im{\\nu}}{2}r\\biggr)}\\alpha_{\\psi,\\mu}^{+}(r)\\;\\text{d}r\\;,\\quad \\mu>1\/4\\;,\\\\\n\t&\\beta^-_{\\psi,\\mu}(R)=\\frac{e^{-\\frac{R}{2}}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{r}{2}}\\sin{\\biggl(\\frac{\\Im{\\nu}}{2}r\\biggr)}\\alpha^{-}_{\\psi,\\mu}(r)\\text{d}r\\;,\\quad\\mu>1\/4\\;,\\\\\n\t&\\beta^{\\pm}_{\\psi,\\mu}(R)=\\frac{e^{-\\frac{1\\mp\\nu}{2}R}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{1\\pm\\nu}{2}r}\\alpha^{\\pm}_{\\psi,\\mu}(r)\\;\\text{d}r\\;,\\quad 0<\\mu<1\/4\\;,\\\\\n\t&\\beta^{+}_{\\psi,1\/4}(R)=\\frac{e^{-\\frac{R}{2}}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{r}{2}}\\alpha^{+}_{\\psi,1\/4}(r)\\;\\text{d}r\\;,\\\\\n\t&\\beta^{-}_{\\psi,1\/4}(R)=\\frac{R^{-1}e^{-\\frac{R}{2}}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}e^{-\\frac{r}{2}}\\alpha^{-}_{\\psi,1\/4}(r)\\;\\text{d}r\\;,\\\\\n\t&\\gamma_{\\psi}(R)=\\frac{e^{-R}}{\\pi(1-2e^{-R}+e^{-2R})}\\int_0^{R}\\int_{S_r}\\mathcal{R}\\psi(\\Gamma,z)\\;\\text{d}\\sigma_r(z)\\;\\text{d}r\n\\end{align*}\n. Because of~\\eqref{eq:integralbounds}, we have the following estimates on the previous coefficients: for any $R\\geq 1$,\n\\begin{align*}\n\t&\\sum_{\\mu\\in \\text{Spec}(\\square)\\cap \\R_{>0}}|\\beta^{\\pm}_{\\psi,\\mu}(R)|\\leq \\frac{5C_{1,s-3}C'_{\\text{Spec}}}{2\\pi}\\norm{\\psi}_{W^{s}}\\;,\\\\\n\t& |\\gamma_{\\psi}(R)|\\leq \\frac{5C_{1,s-3} C_{\\text{Spec}}}{4\\pi} \\norm{\\psi}_{W^{s}}(R+1)e^{-R}\\;,\n\\end{align*}\nusing  the (crude) bound $(1-2e^{-R}+e^{-2R})^{-1}\\leq 5$ in each of the previous inequalities.\n\nThis establishes Proposition~\\eqref{prop:averagedcounting} in its entirety.\n\n\\subsection{Error estimate for the pointwise counting problem}\n\\label{sec:countingproblem}\n\nThis subsection is devoted to the deduction of the estimate on the error for the counting problem stated in Theorem~\\ref{thm:countingproblem}, starting from the asymptotic expansion in~\\eqref{eq:averagedcountingfunction} for the averaged counting function. \n\nRecall from~\\eqref{eq:defaveragedcounting} that, for any real number $R>0$, the ratio $N(R)\/m_{\\mathbb{H}}(B_R)$ equals the value of the function $F_R$ at the identity coset $\\Gamma\\in G\/\\Gamma$. In order to find a convenient approximation for the latter, we shall compare it with the averages \n\\begin{equation*}\n\t\\int_{G\/\\Gamma}\\psi F_R\\;\\text{d}m_{G\/\\Gamma}\n\\end{equation*}  \nwhere the function $\\psi$ ranges over a suitably defined approximate identity\\footnote{The terminology is common in the context of locally compact groups; see, for instance, \\cite[Sec.~2.5]{Folland}.} in $G\/\\Gamma$. \n\nWe now expose the details. Let us fix a parameter $\\delta\\in \\R_{>0}$, on which we shall subsequently impose conditions according to the needs of the argument; choose\n\\begin{itemize}\n\t\\item[(a)] an open symmetric\\footnote{Namely, $U_{\\delta}$ coincides with the set of inverses of its elements.} neighborhood $U_{\\delta}$ of the identity in $G$ such that, for any $R>0$, \n\t\\begin{equation}\n\t\t\\label{eq:neighborhoodcontainment}\n\t\tB_{R-\\delta}\\subset\\bigcap_{g\\in U_{\\delta}}g\\cdot B_R\\subset \\bigcup_{g\\in U_{\\delta}}g\\cdot B_R \\subset B_{R+\\delta}\n\t\\end{equation}\n\t\\item[(b)] and a smooth function $\\psi_{\\delta}\\colon G\/\\Gamma\\to \\R_{\\geq 0}$ with compact support contained in the open set  $U_{\\delta}\\Gamma=\\{g\\Gamma:g\\in U_{\\delta} \\}$ and satisfying\n\t\\begin{equation}\n\t\t\\label{eq:integralone}\n\t\t\\int_{G\/\\Gamma}\\psi_{\\delta}\\;\\text{d}m_{G\/\\Gamma}=1\\;.\n\t\\end{equation}\n\\end{itemize}\n\n\\begin{rmk}\n\tThe existence, for any $\\delta>0$, of a neighborhood $U_{\\delta}$ with the properties claimed above is routinely referred to in the literature (see, for instance,~\\cite{Eskin-McMullen}) as the \\emph{well-roundedness} property of the collection of balls $(B_R)_{R>0}$. A geometric condition of this sort affords to leverage equidistribution results to study lattice point counting problems.\n\\end{rmk}\n\nObserve that we may harmlessly replace $U_{\\delta}$ with\n\\begin{equation*} KU_{\\delta}=\\bigcup_{k\\in K}kU_{\\delta}\\;,\n\\end{equation*}\nand thus assume that $U_{\\delta}$ is saturated with respect to left translations by elements of $K$. Property~\\eqref{eq:neighborhoodcontainment} is unaffected: for any $k\\in K$ and $z\\in \\Hyp$, we have \n\\begin{equation*}\n\td_{\\Hyp}(k\\cdot z,i)=d_{\\Hyp}(k\\cdot z,k\\cdot i)=d_{\\Hyp}(z,i)\\;,\n\\end{equation*}\nas the subgroup $K$ fixes $i$ and acts by hyperbolic isometries; therefore  $k\\cdot B_{r}=B_{r}$ for any $k\\in K$ and any $r>0$. As a consequence of this, we might and shall assume that $\\psi_{\\delta}$ is $K$-invariant.\n\nWe now express, for any $R>0$, the ratio $N(R)\/m_{\\Hyp}(B_R)$ as\n\\begin{equation*}\n\t\\begin{split}\n\t\tF_R(\\Gamma)&=F_R(\\Gamma)-\\int_{G\/\\Gamma}\\psi_{\\delta}F_R\\;\\text{d}m_{G\/\\Gamma}+\\int_{G\/\\Gamma}\\psi_{\\delta}F_R\\;\\text{d}m_{G\/\\Gamma}\\\\\n\t\t&=\\int_{G\/\\Gamma}\\psi_{\\delta}(g\\Gamma)(F_R(\\Gamma)-F_R(g\\Gamma))\\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)+\\int_{G\/\\Gamma}\\psi_{\\delta}F_R\\;\\text{d}m_{G\/\\Gamma}\\;,\n\t\\end{split}\n\\end{equation*}\nwhere the second inequality follows from the property in~\\eqref{eq:integralone}. Let us call $\\mathscr{E}_{\\delta}(R)$, for notational simplicity, the quantity\n\\begin{equation*}\n\t\\int_{G\/\\Gamma}\\psi_{\\delta}(g\\Gamma)(F_R(\\Gamma)-F_R(g\\Gamma))\\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)\\;;\n\\end{equation*}\nin view of~\\eqref{eq:averagedcountingfunction} applied to $\\int_{G\/\\Gamma}\\psi_{\\delta}F_R\\;\\text{d}m_{G\/\\Gamma}$, we may write \n\\begin{equation}\n\t\\label{eq:pointwiseaverage}\n\t\\begin{split}\n\t\tF_R(\\Gamma)=&\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}+\\mathscr{E}_{\\delta}(R)\\\\\n\t\t&+\\text{covol}_{K}(\\Gamma\\cap K)\n\t\t\\biggl( e^{-\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^+_{\\psi_{\\delta},\\mu}(R)+\\beta^-_{\\psi_{\\delta},\\mu}(R)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}R}\\beta^{+}_{\\psi_{\\delta},\\mu}(R)+e^{-\\frac{1-\\nu}{2}R}\\beta^{-}_{\\psi_{\\delta},\\mu}(R)\\\\\n\t\t& +\\varepsilon_0\\biggl(e^{-\\frac{R}{2}}\\beta^{+}_{\\psi_{\\delta},1\/4}(R)+Re^{-\\frac{R}{2}}\\beta^{-}_{\\psi_{\\delta},1\/4}(R)\\biggr)+\\gamma_{\\psi_{\\delta}}(R)\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\nWe estimate, for any $R>0$,\n\\begin{equation}\n\t\\label{eq:Edelta}\n\t|\\mathscr{E}_{\\delta}(R)|\\leq \\int_{G\/\\Gamma}\\psi_{\\delta}(g\\Gamma)|F_R(\\Gamma)-F_R(g\\Gamma)|\\;\\text{d}m_{G\/\\Gamma}(g\\Gamma)\n\t\\leq \\sup_{g\\in U_{\\delta}}|F_R(\\Gamma)-F_R(g\\Gamma)|\\;,\n\\end{equation}\nthe last inequality being a consequence of~\\eqref{eq:integralone} and the fact that $\\text{supp}\\;\\psi_{\\delta}\\subset U_{\\delta}\\Gamma$.\n\nNow, for any $g\\in U_{\\delta}$, we have\n\\begin{equation}\n\t\\label{eq:FR}\n\t\\begin{split}\n\t\t|F_R(\\Gamma)-F_R(g\\Gamma)|&=\\frac{\\bigl||\\Gamma\\cdot H\\cap B_R|-|g\\Gamma\\cdot H\\cap B_R|\\bigr|}{m_{\\mathbb{H}}(B_R)}=\\frac{\\bigl||\\Gamma\\cdot H\\cap B_R|-|\\Gamma\\cdot H\\cap g^{-1}\\cdot B_R|\\bigr|}{m_{\\mathbb{H}}(B_R)}\\\\\n\t\t&\\leq \\frac{\\bigl|\\Gamma\\cdot H\\cap \\bigl(\\bigl(\\bigcup_{g\\in U_{\\delta}}g\\cdot B_R\\bigr)\\setminus \\bigl(\\bigcap_{g\\in U_{\\delta}}g\\cdot B_{R}\\bigr)\\bigr)\\bigr|}{m_{\\mathbb{H}}(B_R)}\\leq\\frac{ N(R+\\delta)-N(R-\\delta)}{m_{\\mathbb{H}}(B_R)}\\\\\n\t\t&=F_{R+\\delta}(\\Gamma)\\frac{m_{\\mathbb{H}}(B_{R+\\delta})}{m_{\\mathbb{H}}(B_R)}-F_{R-\\delta}(\\Gamma)\\frac{m_{\\mathbb{H}}(B_{R-\\delta})}{m_{\\mathbb{H}}(B_R)}\\;,\n\t\\end{split}\n\\end{equation}\nwhere the second-to-last inequality follows from~\\eqref{eq:neighborhoodcontainment}. Choose $R_0=R_0(\\Gamma)>0$ such that the quantities\n\\begin{equation*}\n\tM=\\sup_{r\\geq R_0}F_{r}(\\Gamma) \\quad \\text{ and }\\quad m=\\inf_{r\\geq R_0}F_r(\\Gamma)\n\\end{equation*}\nare non-zero and finite\\footnote{A straightforward modification of the effective argument we are running leads to the well-known non-effective convergence \n\t\\begin{equation*}\n\t\tF_{R}(\\Gamma)\\overset{R\\to\\infty}{\\longrightarrow}\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}\\in \\R_{>0}\\;.\n\t\\end{equation*}\n}. Plugging~\\eqref{eq:FR} into~\\eqref{eq:Edelta}, we get that, for any $R\\geq 2R_0$ and $\\delta<R_0$, \n\\begin{equation}\n\t\\label{eq:neighboringvalues}\n\t\\begin{split}\n\t\t&|\\mathscr{E}_{\\delta}(R)|\\leq (M-m)\\biggl(\\frac{m_{\\mathbb{H}}(B_{R+\\delta})}{m_{\\mathbb{H}}(B_R)}-\\frac{m_{\\mathbb{H}}(B_{R-\\delta})}{m_{\\mathbb{H}}(B_R)}\\biggr)\\\\\n\t\t&=\\frac{(M-m)}{1-2e^{-R}+e^{-2R}}\\biggl(e^{\\delta}(1-2e^{-(R+\\delta)}+e^{-2(R+\\delta)})-e^{-\\delta}(1-2e^{-(R-\\delta)}+e^{-2(R-\\delta)})\\biggr)\\\\\n\t\t&\\leq(M-m)(ce^{\\delta}-e^{-\\delta})\\;,\n\t\\end{split}\n\\end{equation}\nwhere we might for example take $c=c_{\\Gamma}=\\frac{1+e^{-2R_0}}{1-2e^{-R_0}}$.\n\nWe now let the parameter $\\delta$ be a function of the radius $R$; for reasons which we will shortly elucidate (see Remark~\\ref{rmk:exponentialdecay}), we let $\\delta=\\delta(R)=e^{-\\eta R}$ in~\\eqref{eq:pointwiseaverage}, for a certain $\\eta>0$ to be determined later on. In this way, we obtain an expression of the form\n\\begin{equation}\n\t\\label{eq:Rfunctiondelta}\n\t\\begin{split}\n\t\tF_R(\\Gamma)=&\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}+\\mathscr{E}_{e^{-\\eta R}}(R)\\\\\n\t\t&+\\text{covol}_{K}(\\Gamma\\cap K)\n\t\t\\biggl(e^{-\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^+_{\\psi_{e^{-\\eta R}},\\mu}(R)+\\beta^-_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t&+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{-\\frac{1+\\nu}{2}R}\\beta^{+}_{\\psi_{e^{-\\eta R}},\\mu}(R)+e^{-\\frac{1-\\nu}{2}R}\\beta^{-}_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t& +\\varepsilon_0\\biggl(e^{-\\frac{R}{2}}\\beta^{+}_{\\psi_{e^{-\\eta R}},1\/4}(R)+Re^{-\\frac{R}{2}}\\beta^{-}_{\\psi_{e^{-\\eta R}},1\/4}(R)\\biggr)+\\gamma_{\\psi_{e^{-\\eta R}}}(R)\\biggr)\\;,\n\t\\end{split}\n\\end{equation}\nwhich does not depend on the parameter $\\delta$ any longer.\n\nMultiplying by $m_{\\mathbb{H}}(B_R)$ on both sides of~\\eqref{eq:Rfunctiondelta} yields\n\\begin{equation}\n\t\\label{eq:pointwisecounting}\n\t\\begin{split}\n\t\tN(R)=&\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}m_{\\mathbb{H}}(B_R)+\\pi(1-2e^{-R}+e^{-2R})e^{R}\\mathscr{E}_{e^{-\\eta R}}(R)\\\\\n\t\t&+\\pi(1-2e^{-R}+e^{-2R})\\;\\text{covol}_{K}(\\Gamma\\cap K)\n\t\t\\biggl(e^{\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^+_{\\psi_{e^{-\\eta R}},\\mu}(R)+ \\beta^-_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t&\n\t\t+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{\\frac{1-\\nu}{2}R}\\beta^{+}_{\\psi_{e^{-\\eta R}},\\mu}(R)+e^{\\frac{1+\\nu}{2}R}\\beta^{-}_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t& +\\varepsilon_0\\biggl(e^{\\frac{R}{2}}\\beta^{+}_{\\psi_{e^{-\\eta R}},1\/4}(R)+Re^{\\frac{R}{2}}\\beta^{-}_{\\psi_{e^{-\\eta R}},1\/4}(R)\\biggr)+e^{R}\\gamma_{\\psi_{e^{-\\eta R}}}(R)\\biggr)\\;.\n\t\\end{split}\n\\end{equation}\n\nIn order to reach an accurate upper bound for the error\n\\begin{equation*}\n\tE(R)=\\biggl|N(R)-\\frac{\\text{covol}_{K}(\\Gamma\\cap K)}{\\text{covol}_{G}(\\Gamma)}m_{\\mathbb{H}}(B_R)\\biggr|\\;,\n\\end{equation*}\nin our counting problem, it remains to determine which are the highest-order terms in the expansion~\\eqref{eq:pointwisecounting}. To this end, it is relevant to estimate the Sobolev norms of the functions $\\psi_{e^{-\\eta R}}$ for $R>0$, because of the bounds in~\\eqref{eq:betabound} and~\\eqref{eq:gammabound}.\n\n\n\\begin{lem}\n\t\\label{lem:growthSobolev}\n\tFor any $0<\\delta<1$, the function $\\psi_{\\delta}$ can be chosen to satisfy \n\t\\begin{equation}\n\t\t\\label{eq:decaySobolev}\n\t\t\\norm{\\psi_{\\delta}}_{W^{s}}\\leq \\delta^{-(1+s)}\\norm{\\psi_1}_{W^{s}}\n\t\\end{equation}\n\tfor any $s>0$.\n\\end{lem}\n\\begin{proof}\n\tRecall that $\\psi_\\delta$ is assumed to be $K$-invariant or, in other words, a smooth compactly supported function on the two-dimensional manifold $K\\bsl G\/\\Gamma$. Since any Riemannian metric on $K\\bsl G\/\\Gamma$ is equivalent, on a fixed compact coordinate ball containing the identity coset $Ke\\Gamma$, to the Euclidean metric on a compact neighborhood of the origin in $\\R^{2}$ (cf.~\\cite[Lem.~13.28]{Lee}), the problem of constructing $\\psi_{\\delta}$ so to meet our requirement can be transferred to the Euclidean plane. Specifically, we would like to construct a collection $(\\psi_{\\delta})_{0<\\delta\\leq1}$ of mollifiers (cf.~\\cite[Sec.~4.4]{Brezis}) so that~\\eqref{eq:decaySobolev} is satisfied, where $\\norm{\\cdot}_{W^{s}}$ are now the standard fractional Sobolev norms on $\\R^{2}$. A straightforward computation allows to ascertain that the customary choice \n\t\\begin{equation*}\n\t\t\\psi_{\\delta}(x)=\\frac{1}{\\delta^2}\\psi_1\\biggl(\\frac{x}{\\delta}\\biggr)\\;,\\quad x\\in \\R^{2},\n\t\\end{equation*}\n\twhere $\\psi_1$ is a fixed compactly supported smooth nonnegative function with unit average over $\\R^{2}$, fulfills~\\eqref{eq:decaySobolev}. \n\\end{proof}\n\nHenceforth, we assume that the collection $(\\psi_{\\delta})_{0<\\delta<1}$ satisfies the condition in Lemma~\\ref{lem:growthSobolev}.\n\nWe remind the reader that we indicate with $\\mu_*$ the spectral gap of the hyperbolic surface $S=\\Gamma\\bsl \\Hyp$, that is, the infimum of the set $\\text{Spec}(\\square)\\cap \\R_{>0}$. Also, we denote by $\\nu_*$ the complex number defined by the properties $\\nu_*\\in \\R_{\\geq 0}\\cup i\\R_{>0}$ and $1-\\nu_*^2=4\\mu_*$.\n\nSince $e^{\\delta}-e^{-\\delta}\\sim 2\\delta$ for $\\delta\\sim 0$, we deduce from~\\eqref{eq:neighboringvalues} that the term $e^{R}\\mathscr{E}_{e^{-\\eta R}}(R)$ is at most of order $e^{(1-\\eta)R}$. On account of Lemma~\\ref{lem:growthSobolev}, the highest-order term in the expression \n\\begin{equation*}\n\t\\begin{split}\n\t\t&e^{\\frac{R}{2}} \\sum_{\\mu\\in \\text{Spec}(\\square),\\;\\mu>1\/4}\\beta^+_{\\psi_{e^{-\\eta R}},\\mu}(R)+ \\beta^-_{\\psi_{e^{-\\eta R}},\\mu}(R)\n\t\t+\\sum_{\\mu\\in \\text{Spec}(\\square),\\;0<\\mu<1\/4}e^{\\frac{1-\\nu}{2}R}\\beta^{+}_{\\psi_{e^{-\\eta R}},\\mu}(R)\\\\\n\t\t& +e^{\\frac{1+\\nu}{2}R}\\beta^{-}_{\\psi_{e^{-\\eta R}},\\mu}(R)+\\varepsilon_0\\biggl(e^{\\frac{R}{2}}\\beta^{+}_{\\psi_{e^{-\\eta R}},1\/4}(R)+Re^{\\frac{R}{2}}\\beta^{-}_{\\psi_{e^{-\\eta R}},1\/4}(R)\\biggr)+e^{R}\\gamma_{\\psi_{e^{-\\eta R}}}(R)\n\t\\end{split}\n\\end{equation*}\nis $e^{\\frac{1+\\Re{\\nu_*}}{2}R}\\beta_{\\psi_{e^{-\\eta R}},\\mu_*}(R)$; because of Lemma~\\ref{lem:growthSobolev} and~\\eqref{eq:betabound}, the latter is at most of order\n\\begin{equation*}\n\te^{\\frac{1+\\Re{\\nu_*}}{2}R}e^{(1+s)\\eta R}=e^{\\frac{1+\\Re{\\nu_*}+2(1+s)\\eta}{2}R}\\;.\n\\end{equation*}\n\n\\begin{rmk}\n\t\\label{rmk:exponentialdecay}\n\tThe reason for choosing $\\delta$ to decay exponentially fast with $R$ becomes now apparent: it is the only way to get a sensible comparison between the orders of the two terms considered above. \t\n\\end{rmk}\n\nOstensibly the optimal choice of the parameter $\\eta$ for our purposes is\n\\begin{equation*} \\eta=\\frac{1-\\Re{\\nu_*}}{2(2+s)}\\;,\n\\end{equation*}\nwhich realizes the equality of exponents\n\\begin{equation*}\n\t1-\\eta=\\frac{1+\\Re{\\nu_*}+2(1+s)\\eta}{2}\\;.\n\\end{equation*}\nBearing in mind that the $K$-invariance of $\\psi_{e^{-\\eta R}}$ allows to choose $s$ can arbitrarily close to $9\/2$ (cf.~Theorem~\\ref{thm:expandingonsurface}), it is straightforward to deduce that, setting $\\eta_*=\\frac{1}{13}(1-\\Re{\\nu_{*}})$,  we have\n\\begin{equation*}\n\t\\lim\\limits_{R\\to\\infty}\\frac{E(R)}{e^{(1-\\eta_{*}+\\varepsilon)R}}=0\n\\end{equation*} \nfor any $\\varepsilon>0$, which establishes Theorem~\\ref{thm:countingproblem}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\footnotesize\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\sectlabel{intro}\n\nAmong the many exciting new applications of quantum physics\nin the realm of computation and information theory,\nI~am particularly fond of quantum cryptography,\nquantum computing and quantum teleportation~\\cite{LNCS}.\nQuantum cryptography allows for the confidential transmission\nof classical information under the nose of an eavesdropper,\nregardless of her computing power or technological\nsophistication~\\cite{BB84,BBBSS,sciam}.   Quantum computing allows for an\nexponential amount of computation to take place simultaneously in a single\npiece of hardware~\\cite{feynman,deutsch}; \nof~particular interest is the ability of quantum computers to factorize\nnumbers very efficiently~\\cite{shor}, with dramatic implications for\nclassical cryptography~\\cite{RSA}.  Quantum teleportation allows for the\ntransmission of quantum information to a distant location despite the\nimpossibility of measuring or broadcasting the information to be\ntransmitted~\\cite{BBCJPW}. Each of these concepts had a strong overtone of\nscience fiction when they were first introduced.\n\nIf~asked to rank these ideas on a scale of technological\ndifficulty, it is tempting to think that quantum cryptography\nis easiest while quantum teleportation is the most outrageous---especially\nwhen it comes to teleporting goulash~\\cite{IBM}!\nThis ranking is correct with respect to quantum cryptography,\nwhose feasibility has been demonstrated by several experimental\nprototypes capable of reliably transmitting confidential information\nover distances of tens of kilometres~\\cite{townsend,gisin,hughes}.\nThe situation is less clear when it comes to comparing the technological\nfeasibility of quantum computing with that of quantum teleportation.\n\nOn~the one hand, quantum teleportation can be implemented with a quantum\ncircuit that is much simpler than that required by any\nnontrivial quantum computational task: the~state of an\narbitrary qubit (quantum bit) can be teleported with as few as two quantum\nexclusive-or (controlled-not) gates.  Thus, quantum teleportation\nis significantly easier to implement than quantum computing if we\nare concerned only with the complexity of the required circuitry.\n\nOn~the other hand, quantum computing is meaningful\neven if it takes place very quickly---indeed its primary purpose\nis increased computational speed---and within a small region of space.\nQuite the opposite,\nthe interest of quantum teleportation would be greatly reduced if the actual\nteleportation had to take place immediately after the required preparation.\nThus, a working demonstration of quantum teleportation is likely to be seen\nbefore the quantum factorization of even a very small integer is achieved, but\nquantum teleportation across significant time and space will have to await a\ntechnology that allows for the efficient long-term storage of quantum\ninformation.\nNevertheless, it may be that short-distance quantum teleportation\nwill play a role in transporting quantum information inside quantum\ncomputers.  Thus we see that the fates of quantum computing and\nquantum teleportation are entangled!\n\n\\section{Quantum teleportation}\\sectlabel{teleport}\n\nRecall that any attempt at measuring quantum information\ndisturbs it irreversibly and yields incomplete information.\nThis makes it impossible to transmit quantum information\nthrough a classical channel.  Recall also that the purpose\nof quantum teleportation~\\cite{BBCJPW} is to circumvent this\nimpossibility so as to allow the faithful transmission of\nquantum information between two parties, conventionally\nreferred to as Alice and Bob.\n\nIn~order to achieve teleportation, Alice and Bob must\nshare prior quantum entanglement.  This is usually explained in terms\nof Einstein--Podolsky--Rosen nonlocal quantum states~\\cite{EPR}\nand Bell measurements, which makes the process seem very mysterious.\nThe~purpose of this note is to show how to achieve quantum teleportation\nvery simply in terms of quantum computation.  As~interesting side\nproduct, we obtain a quantum circuit with the unusual feature that\nthere are points in the circuit at which the quantum information can\nbe completely disrupted by a measurement---or~some types of interaction\nwith the environment---without ill effects:\nthe same final result is obtained whether or not measurement takes place.\nThis is true despite that fact that the qubits affected by these\nmeasurements are entangled with the other qubits carried by the\ncircuit, which should make these measurements even more damaging.\n\nOf~course, the uncanny power of quantum computation draws in parts\non nonlocal effects inherent to quantum mechanics.\nThe~quantum teleportation circuit described in~\\sect{circuit} is not really\ndifferent in principle from the original idea~\\cite{BBCJPW} since\nit uses quantum computation to create and measure nonlocal states.\nNevertheless it sheds new light on teleportation, at least from a\npedagogical point of view, since it makes the process completely\nstraightforward to anyone who believes that quantum computation is\na reasonable proposition.  Moreover, this circuit could genuinely be\nused for teleportation purposes inside a quantum computer.\nFinally, the surprising resilience of this circuit to measurements\nperformed while it is processing information\nmay turn out to have relevance to quantum error correction. \n\n\\section{The basic ingredients}\\sectlabel{ingredients}\n\nAs is often the case with quantum computation, we shall need two basic\ningredients: the exclusive-or gate (also known as controlled-not), which\nacts on two qubits at once, and arbitrary unitary operations on single qubits.\nLet~\\ket{0} and~\\ket{1} denote basis states for single qubits and recall\nthat pure states are given by linear combination of basis states\nsuch as \\mbox{$\\ket{\\psi}=\\alpha\\ket{0}+\\beta\\ket{1}$} where $\\alpha$ and\n$\\beta$ are complex numbers such that \\mbox{$\\norm{\\alpha}+\\norm{\\beta}=1$}.\n\nThe quantum exclusive-or (XOR), denoted as follows,\n\\begin{center}\n\\begin{picture}(180,60)\n\\thicklines\n\\put(0,40){\\makebox(20,20){\\sf a}}\n\\put(25,50){\\line(1,0){130}}\n\\put(90,50){\\circle*{10}}\n\\put(160,40){\\makebox(20,20){\\sf x}}\n\\put(0,0){\\makebox(20,20){\\sf b}}\n\\put(25,10){\\line(1,0){130}}\n\\put(90,10){\\circle{20}}\n\\put(160,0){\\makebox(20,20){\\sf y}}\n\\put(90,0){\\line(0,1){50}}\n\\end{picture}\n\\end{center}\nsends \\ket{00} to \\ket{00}, \\ket{01} to \\ket{01}, \\ket{10} to \\ket{11}\nand \\ket{11} to~\\ket{10}.  In~other words,\n{\\em provided the input states at {\\sf a} and {\\sf b} are in basis states},\nthe output state at {\\sf x} is the same as the input state at {\\sf a},\nand the output state at {\\sf y} is the exclusive-or of the two input states\nat {\\sf a} and~{\\sf b}.  This is also known as the controlled-not gate because\nthe state carried by the {\\em control} wire ``{\\sf ax}'' is not disturbed\nwhereas the state carried by the {\\em controlled} wire ``{\\sf by}'' is\nflipped if and only if the state on the control wire was~\\ket{1}.\nNote that the classical interpretation given above no longer holds \nif the input qubits are not in basis states: it is possible for\nthe output state on the control wire (at~{\\sf x}) to be different from its\ninput state (at~{\\sf a}).  Moreover, the joint state of the output\nqubits can be entangled even if the input qubits were not, and\nvice versa.\n\nIn addition to the quantum exclusive-or, we shall need two single-qubit\nrotations {\\sf L} and {\\sf R}, and two single-qubit conditional phase-shifts\n{\\sf S} and~{\\sf T}\\@.  Rotation {\\sf L} sends \\ket{0} to\n\\mbox{$(\\ket{0}+\\ket{1})\/\\sqrt{2}$} and \\ket{1}\nto \\mbox{$(-\\ket{0}+\\ket{1})\/\\sqrt{2}$},\nwhereas {\\sf R}~sends \\ket{0} to\n\\mbox{$(\\ket{0}-\\ket{1})\/\\sqrt{2}$} and \\ket{1}\nto \\mbox{$(\\ket{0}+\\ket{1})\/\\sqrt{2}$}.\nNote that \\mbox{$\\mbox{\\sf LR}\\ket{\\psi}=\\mbox{\\sf RL}\\ket{\\psi}=\\ket{\\psi}$}\nfor any qubit~\\ket{\\psi}.\nConditional phase-shift {\\sf S} sends \\ket{0} to $i\\ket{0}$\nand leaves \\ket{1} undisturbed, whereas {\\sf T} sends\n\\ket{0} to $-\\ket{0}$ and \\ket{1} to $-i\\ket{1}$.\nIn~terms of unitary matrices, the operations are\n\\[\n\\begin{array}{lll}\\multicolumn{2}{l}{\n\\mbox{\\sf L} = {\\displaystyle \\frac{1}{\\sqrt{2}}}\n\\left( \\begin{array}{rr}1&-1\\\\1&1\\end{array} \\right)} &\n\\mbox{\\sf R} = {\\displaystyle \\frac{1}{\\sqrt{2}}}\n\\left( \\begin{array}{rr}1&1\\\\-1&1\\end{array} \\right) \\\\[7mm]\n\\mbox{\\sf S} = \\left( \\begin{array}{rr}i&0\\\\0&1\\end{array} \\right)\n& \\mbox{and} &\n\\mbox{\\sf T} = \\left( \\begin{array}{rr}-1&0\\\\0&-i\\end{array} \\right)\n\\end{array}\n\\]\nif $\\alpha\\ket{0}+\\beta\\ket{1}$ is represented by vector \\state{\\alpha}{\\beta}.\nSimilarly the quantum exclusive-or operation is given by matrix\n\\[\\mbox{\\sf XOR} = \\left(\n\\begin{array}{llll}1&0&0&0\\\\0&1&0&0\\\\0&0&0&1\\\\0&0&1&0\\end{array}\\right)\\]\nif \\mbox{$\\alpha\\ket{00}+\\beta\\ket{01}+\\gamma\\ket{10}+\\delta\\ket{11}$} is\nrepresented by the transpose of vector \\mbox{$(\\alpha,\\beta,\\gamma,\\delta)$}.\n\n\\section{The teleportation circuit}\\sectlabel{circuit}\n\nConsider the following quantum circuit.  Please disregard the dashed line\nfor the moment.\n\\begin{center}\n\\begin{picture}(495,210)(0,-40)\n\\thicklines\n\\put(0,135){\\makebox(20,20){\\sf a}}\n\\put(25,145){\\line(1,0){155}}\n\\put(150,145){\\circle*{10}}\n\\put(180,130){\\framebox(30,30){\\sf R}}\n\\put(210,145){\\line(1,0){60}}\n\\put(270,130){\\framebox(30,30){\\sf S}}\n\\put(300,145){\\line(1,0){70}}\n\\put(335,145){\\circle{20}}\n\\put(370,130){\\framebox(30,30){\\sf S}}\n\\put(400,145){\\line(1,0){70}}\n\\put(435,145){\\circle{20}}\n\\put(475,135){\\makebox(20,20){\\sf x}}\n\\put(0,70){\\makebox(20,20){\\sf b}}\n\\put(25,80){\\line(1,0){25}}\n\\put(50,65){\\framebox(30,30){\\sf L}}\n\\put(80,80){\\line(1,0){390}}\n\\put(110,80){\\circle*{10}}\n\\put(150,80){\\circle{20}}\n\\put(285,80){\\circle*{10}}\n\\put(475,70){\\makebox(20,20){\\sf y}}\n\\put(0,5){\\makebox(20,20){\\sf c}}\n\\put(25,15){\\line(1,0){345}}\n\\put(110,15){\\circle{20}}\n\\put(285,15){\\circle{20}}\n\\put(335,15){\\circle*{10}}\n\\put(370,0){\\framebox(30,30){\\sf T}}\n\\put(400,15){\\line(1,0){70}}\n\\put(435,15){\\circle*{10}}\n\\put(110,5){\\line(0,1){75}}\n\\put(150,70){\\line(0,1){75}}\n\\put(285,5){\\line(0,1){75}}\n\\put(335,15){\\line(0,1){140}}\n\\put(435,15){\\line(0,1){140}}\n\\put(475,5){\\makebox(20,20){\\sf z}}\n\\thinlines\n\\put(240,-40){\\dashbox{7.5}(0,210){}}\n\\put(25,-40){\\makebox(215,30){\\sl Alice}}\n\\put(240,-40){\\makebox(230,30){\\sl Bob}}\n\\end{picture}\n\\end{center}\nLet \\ket{\\psi} be an arbitrary one-qubit state.  Consider\nwhat happens if you feed \\ket{\\psi00} in this circuit,\nthat is if you set upper input {\\sf a} to \\ket{\\psi} and both other\ninputs {\\sf b} and {\\sf c} to~\\ket{0}.  It~is a straightforward\nexercise to verify that state \\ket{\\psi} will be transferred\nto the lower output~{\\sf z}, whereas both other outputs {\\sf x} and {\\sf y}\nwill come out in state \\mbox{$\\ket{\\phi}=(\\ket{0}+\\ket{1})\/\\sqrt{2}$}.\nIn~other words the output will be~\\ket{\\phi\\phi\\psi}.\nIf~the two upper outputs are measured in the standard basis\n\\mbox{(\\ket{0} versus \\ket{1})}, two random classical bits will be obtained\nin addition to quantum state \\ket{\\psi} on the lower \\mbox{output}.\n\nNow, let us consider the state of the system at the dashed line.\nA~simple calculation shows that all three qubits are entangled.\nWe~should therefore be especially careful not to\ndisturb the system at that point.  Never\\-the\\-less, let us \nmeasure the two upper qubits, leaving the lower qubit undisturbed.\nThis~measurement results in two purely random classical bits $u$ and~$v$,\nbearing no correlation whatsoever with the original state~\\ket{\\psi}.\nLet~us now turn $u$ and $v$ back into quantum bits and reinject\n\\ket{u} and \\ket{v} in the circuit immediately after the dashed line.\n\nNeedless to say that the quantum state carried at the dashed line\nhas been completely disrupted by this measurement-and-resend process.\nWe~would therefore expect this disturbance to play havoc with the\nfinal output of the circuit.  Not~at all!  In~the end, the state\ncarried at {\\sf xyz} is~\\ket{uv\\psi}.  In~other words, \\ket{\\psi}\nis still obtained at~{\\sf z} and the other two qubits, if measured,\nare purely random provided we forget the measurement outcomes at the\ndashed line.  Another way of seeing this phenomenon is that the\noutcome of the circuit will not be altered if the state of the upper\ntwo qubits leaks to the environment (in the standard basis)\nat the dashed line.\n\nTo~turn this circuit into a quantum teleportation \\mbox{device}, we need\nthe ability to store qubits.  Assume Alice prepares two qubits in\nstate \\ket{0} and pushes them through the first two gates of the\ncircuit.\n\\begin{center}\n\\begin{picture}(325,80)(0,5)\n\\thicklines\n\\put(0,60){\\makebox(20,20){\\ket{0}}}\n\\put(25,70){\\line(1,0){25}}\n\\put(50,55){\\framebox(30,30){\\sf L}}\n\\put(80,70){\\line(1,0){75}}\n\\put(160,60){\\makebox(20,20){$\\sigma$}}\n\\put(110,70){\\circle*{10}}\n\\put(0,5){\\makebox(20,20){\\ket{0}}}\n\\put(25,15){\\line(1,0){300}}\n\\put(110,15){\\circle{20}}\n\\put(330,5){\\makebox(20,20){$\\rho$}}\n\\put(110,5){\\line(0,1){65}}\n\\end{picture}\n\\end{center}\nShe keeps the upper qubit $\\sigma$ in quantum memory and gives the other,\n$\\rho$, to Bob.  [We~do not denote these qubits by kets because\nthey are not individual pure states: \\mbox{together} they are in state\n\\mbox{$\\Phi^{+}=(\\ket{00}+\\ket{11})\/\\sqrt{2}$}.]\nAt~some later time, Alice receives a mystery qubit in \\mbox{unknown}\nstate~\\ket{\\psi}. In~order to teleport this qubit to Bob, she releases\n$\\sigma$ from her quantum memory and pushes it together with the mystery\nqubit through the next two gates of the circuit.  She measures both\noutput wires to turn them into classical bits $u$ and~$v$.\n\\begin{center}\n\\begin{picture}(200,80)(75,70)\n\\thicklines\n\\put(75,125){\\makebox(20,20){\\ket{\\psi}}}\n\\put(100,135){\\line(1,0){80}}\n\\put(150,135){\\circle*{10}}\n\\put(180,120){\\framebox(30,30){\\sf R}}\n\\put(210,135){\\line(1,0){40}}\n\\put(255,125){\\makebox(20,20){$u$}}\n\\put(75,70){\\makebox(20,20){$\\sigma$}}\n\\put(100,80){\\line(1,0){150}}\n\\put(150,80){\\circle{20}}\n\\put(255,70){\\makebox(20,20){$v$}}\n\\put(150,70){\\line(0,1){65}}\n\\end{picture}\n\\end{center}\nTo complete teleportation, Alice has to communicate $u$ and $v$ to Bob\nby way of a classical communication channel.  Upon reception of the\nsignal, Bob creates quantum states \\ket{u} and \\ket{v} from the\nclassical information received from Alice, he releases the qubit\n$\\rho$ he had kept in quantum memory, and he pushes all three\nqubits into his part of the circuit (on~the right of the dashed line).\nFinally Bob may wish to measure the two upper qubit at {\\sf x} and~{\\sf y}\nto make sure that he gets $u$ and~$v$; otherwise something went wrong\nin the teleportation apparatus.  At~this point, teleportation is\ncomplete as Bob's output {\\sf z} is in state~\\ket{\\psi}.\nNote that this process works equally well if Alice's mystery qubit\nis not in a pure state.  In~particular, Alice can teleport to Bob\nentanglement with an arbitrary auxiliary system, possibly outside both\nAlice's and Bob's labo\\-ra\\-tories.\n\nIn~practice, Bob need not use the quantum circuit shown right of\nthe dashed line at~all.  Instead, he may choose classically one of 4\npossible rotations to apply to the qubit he had kept in quantum memory,\ndepending on the 2 classical bits he receives from Alice.\n(This would be more in tune with the original teleportation\nproposal~\\cite{BBCJPW}.)  This explains the earlier claim\nthat quantum teleportation can be achieved at the cost of\nonly two quantum exclusive-ors: those of Alice.\nNever\\-theless, the unitary version of Bob's process\ngiven here may be more appealing than choosing classically\namong 4 courses of action if teleportation is used\ninside a quantum computer.\n\n\n\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nComposite supernova remnants (SNRs) are a subclass of core collapse SNRs where non-thermal radio emission is observed from both the expanding shell of the SNR, and from the PWN located inside it. The PWNe are powered by the relativistic outflows from the young neutron star and can be observed at all wavelengths, but mostly in the radio and X-rays.\nThe PWN morphology can provide crucial information on the properties of the outflow, the interacting ambient medium and the geometry of the pulsar powering it \\citep[see][and references therein for detailed understanding of the structure and evolution of PWNe]{gaensler2006}. The {\\it Chandra~\\\/} observatory with its excellent spatial resolution and high sensitivity has provided significant breakthroughs in the study of PWNe. Apart from steadily increasing the census on the number of PWNe discovered, it has provided a unique opportunity to study in detail their spatial and spectral structures and signatures of interaction with the surrounding medium \\citep{pwnchandra}. Most of these are however Galactic sources and there are no confirmed PWNe in our satellite galaxy the SMC, yet. To this day a total of 24 classified SNRs are known in the SMC \\citep[][and references therein]{haberl2012}. Out of them very few are PWN candidates. One such source, HFPK 334 has been recently dismissed as a background source \\citep{crawford2014}. Discovery of a PWN in the SMC will open a new window in the study of rotation powered pulsars in the satellite galaxy which has a rich history of active star formation and is therefore expected to host young energetic pulsars.\n\nIKT 16 is an X-ray and radio-faint SNR in the SMC first studied with {\\it XMM-Newton~\\\/} in a survey of known SNRs in the SMC \\citep{van2004}. In this study a region of harder \nX-ray emission was found at the centre of the remnant although it could not be probed further  due to poor statistics with a single exposure. \\cite{owen2011} further used eight more archival {\\it XMM-Newton~\\\/} {observations~\\\/} \n(total useful exposure 125 ks and off-axis angle 8-12$\\arcmin$) taken subsequently to carry out a multiwavelength study of the spatial and spectral properties of this SNR and the associated central source. The authors found substantial evidence that the unresolved source detected at the centre of the SNR by XMM-Newton, is a PWN associated with it.\nRadio images from the ATCA and MOST surveys displayed faint radio structures correlated with the X-ray throughout the remnant. The brightest feature was an extended radio emission \ncorresponding to the X-ray source and extending a distance of 40\\hbox{$^{\\prime\\prime}$}~ towards the centre of the SNR \\citep{owen2011}. This picture is consistent with a moving pulsar in the SNR.\n Further, the large size of the remnant suggested that the SNR is in the adiabatic Sedov phase of evolution. In the Sedov model the age of the SNR was $\\sim$ 14.7 kyr\nand implied the PWN may have interacted with and been compressed by the reverse shock. The bright central source was located 30\\hbox{$^{\\prime\\prime}$}~ from the SNR centre implying a transverse kick \nvelocity of $\\sim$ 580 km s$^{-1}$. \n\nTo further resolve the nature of the central source and establish the presence of the first PWN in the SMC, a 40 ks on-axis {\\it Chandra~\\\/} observation was solicited. In this paper we describe the results of the {\\it Chandra~\\\/} {observation~\\\/} of IKT 16. We have performed detailed spatial and spectral analysis of the hard X-ray emission near the centre of the SNR, previously unresolved. With the unprecedented spatial resolution of {\\it Chandra~\\\/} we have resolved the source into a central point source (a putative pulsar) and an extended emission surrounding it, indicating a PWN nature for the source. The {observations~\\\/} and analysis are described in section 2. Section 3 presents the spatial analysis including the imaging and morphological fitting of the source. Section  4 presents the detailed spectral analysis of the central source, the surrounding nebula and its decomposed regions. Section 5 presents the discussion  and section 6 the conclusions.\n\n\\begin{figure*}\n\\hspace*{-0.52cm}\n\\subfigure[]{\\includegraphics[angle=0,scale=0.5]{src-col-new.ps}}\n\\hspace*{-1.5cm}\n\\subfigure[]{\\includegraphics[angle=0,scale=0.45]{model-new-contour.ps}}\n\\subfigure[]{\\includegraphics[angle=0,scale=0.52]{src_region-col-new.ps}}\n\\vspace*{2.0cm}\n\\hspace*{0.18 cm}\n\\subfigure[]{\\includegraphics[angle=0,scale=0.42]{new_extended_smoothed_image.ps}}\n\\caption{a({\\it top left}): {\\it Chandra~\\\/} ACIS-S (0.5--8 keV) full resolution image of the PWN in IKT 16. The image size is 20\\hbox{$^{\\prime\\prime}$} x 25\\hbox{$^{\\prime\\prime}$}. The scale is in square root, the X and Y axis in degrees, and units are counts for all the images. The brightest pixel in the centre corresponds to the putative pulsar. b({\\it top right}): Best-fit model of the PWN (0.5--8 keV). Overlayed are contours from the data which have been smoothed with a Gaussian of $\\sigma = 3$\\hbox{$^{\\prime\\prime}$}. The contour levels are plotted at values of 0.7, 1.6 and 30 counts arcsecs$^{-2}$. \nc({\\it bottom left}): Same as the {\\it top left} figure showing the regions used for spectral extraction. The central circle corresponds to the pulsar, and the rectangular box the entire nebula with the central point source removed. The dashed circle corresponds to the outer boundary of the inner, and the inner boundary of the outer nebular extraction regions respectively. d({\\it bottom right}): Larger (0.5--2 keV) image centred on the PWN of IKT 16 (ACIS-S3), with the bottom right corner of the image corresponding to ACIS-S2. The box region used for nebular extraction is shown in black solid lines and the background annular region used for spectral extraction (inner radius 34\\hbox{$^{\\prime\\prime}$}) is shown in white solid lines. The white dashed circle indicates the position and extent of the SNR in \\cite{owen2011}. The point source contribution at the centre has been removed and replaced with values obtained by interpolation from the nebular region. The image has been smoothed using a Gaussian kernel of width 5\\hbox{$^{\\prime\\prime}$}.}\n\\label{images}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.62,angle=-90]{ikt16-new.eps}\n\\caption{ATCA 2.1\\,GHz image of PWN IKT\\,16 overlaid with Chandra contours (\\textbf{1.25, 5, 25 and 100 c\/arcsec$^{2}$}). The cross marked in green denotes the centre of the SNR in IKT 16.\nThe synthesis beam size of $3.38\\arcsec \\times 3.16\\arcsec$ at PA of $-17.8^{\\circ}$ is shown on the bottom left corner. The 2.1\\,GHz image grayscale bar is shown in units of mJy\/beam.}\n\\label{radio}\n\\end{figure*}\n\n\n\\section{Observations \\& Analysis}\n\\label{sec:obs}\n\n\\subsection{{\\it Chandra~\\\/} ACIS {observation~\\\/} and data reduction}\n\\label{sec:xobs}\n\nThe {\\it Chandra~\\\/} {observation~\\\/} (ObsID 13773) was carried out with the ACIS-S as the primary instrument in the timed exposure mode. The {observation~\\\/} was performed on 09-02-2013 with the central source of  IKT 16 positioned at the aimpoint of the S3 CCD with an exposure of 38.5 ks. \nIt is worthwhile to mention here that there was a previous {\\it Chandra~\\\/} {observation~\\\/} (obsID 2948, 9 ks) where the source was detected \\citep{evans2010}. But the short duration of this {observation~\\\/} and its large off axis\nangle prevented further investigation of the source from this data. Data reduction was performed using CIAO 4.6 using CALDB 4.6.3 and the standard analysis procedure prescribed \\footnote{http:\/\/cxc.harvard.edu\/ciao\/}.\nThe level 1 event file was reprocessed with \\textit{chandra$\\_$repro}, which incorporates the subpixel repositioning algorithm EDSER as a default for attaining better angular resolution for sources near the centre of the FOV. The effective exposure of the observation after filtering was 38.5 ks. We also checked for possible presence of pileup in the data. Using the tool {\\it pileup$\\_$map}, the estimated fraction of pileup in the centremost pixel is < $5\\%$. Hence the effect is not important for our {observation~\\\/}.\n\nImages were created using the task \\textit{dmcopy}. Spatial analysis was performed using the \\textit{Sherpa} analysis package 4.4 \\footnote{http:\/\/cxc.harvard.edu\/sherpa4.4\/}. The task {\\it specextract} was used for extracting source and background spectra and response files from regions of interest.\nThe selection of regions used for spectral extraction is described in the spectral analysis section. Spectra were fitted using \\textit{XSPEC v12.8.1} \\footnote{http:\/\/heasarc.gsfc.nasa.gov\/xanadu\/xspec\/}.\n \n\\subsection{Imaging}\n\\label{sec-im-an}\nFigure \\ref{images}a shows a full resolution {\\it Chandra~\\\/} image (0.5--8 keV) centred on the source near the centre of IKT 16, and zoomed into a region 20\\hbox{$^{\\prime\\prime}$} x 25\\hbox{$^{\\prime\\prime}$}~ wide. The source appears to be symmetrically elongated in the east-west direction with a bright point source located at its centre. The elongation measures about 5\\hbox{$^{\\prime\\prime}$}.\nTo accurately determine the position of the central source, we created a subpixel image of the same (at $\\frac{1}{5}$ of the ACIS pixel resolution) and applied the source detection algorithm {\\it celldetect}. The coordinates of the point source are RA(J2000)=$00^h58^m16.85^s$ Dec=$-72\\deg18\\arcm05.60\\hbox{$^{\\prime\\prime}$}$ considering an error of 0.6\\hbox{$^{\\prime\\prime}$}~ at 90 $\\%$ confidence level in absolute {\\it Chandra~\\\/} astrometry. The net count rate from the point source after subtracting the nebular component is 0.011 c\/s, and from the entire nebula after subtracting the point source contribution is 0.003 c\/s. The details of background subtraction are discussed in the spectral analysis section \\ref{sec-spec}. The point source is about three times\nbrighter than the extended emission. \nAfter removing the contribution from the bright point source at the centre, and smoothing the image with a Gaussian kernel of width 5\\arcsec, there is evidence of a diffuse emission extending further out up to $\\sim$ 30$\\hbox{$^{\\prime\\prime}$}$ (see Fig.~\\ref{images}d). This diffuse component is discussed in Sect.\\ref{diff-neb}. The same image shows hints of the presence of the SNR, especially the excess in the north coincident with that in the ${\\it XMM-Newton~\\\/}$ image of IKT 16 \\citep{owen2011}.\n\n\n\\subsection{Radio observations}\n\\label{sec:radioobs}\n\nOur new ATCA observations, project C2521 (CI: J. van Loon), used the Compact Array Broadband Backend (CABB) with the 6A array configuration at 2.1\\,GHz providing improved flux density estimates and resolution. These new images were acquired on 02-01-2012 with $\\sim10.48$~hours integration over the 12~hour observing session. The radio galaxy PKS 1934-638 was used as a primary flux calibration source for all observations, with the radio sources PKS 0230-790 and PKS 2353-686 used for phase calibration. A standard calibration process was carried out using the \\textsc{miriad} data reduction software package (Sault et al. 2011). In order to improve the fidelity and sensitivity of the final image, a single iteration of self-calibration was performed on the strongest sources in the field. A uniform weighting scheme was subsequently used throughout the imaging process, as it provided a balance between improving the theoretical rms noise while maintaining an adequate beam shape. Given the 2\\,GHz of bandwidth provided by CABB, images were formed using \\textsc{miriad} multi-frequency synthesis (\\textsc{mfclean}; Sault \\& Wieringa 1994). The same procedure was used for both {\\it U} and {\\it Q} Stokes parameter maps. However, there was no reliable detection in the {\\it U} or {\\it Q} intensity parameters associated with this object, implying a lack of polarisation at lower radio frequencies (2.1\\,GHz). The final image produced (after primary beam corrected), which possess a FWHM of $3.38\\arcsec \\times 3.16\\arcsec$ and a PA of $-17.8^{\\circ}$, has a 1$\\sigma$ rms noise level of 16\\,$\\mu$Jy. In Fig.~\\ref{radio} we show the ATCA 2.1\\,GHz surface brightness image of PWN IKT\\,16.\n   \n\\section{X-ray Spatial analysis}\n\\label{sec:xspec}\nTaking advantage of the excellent spatial resolution of {\\it Chandra}, we performed detailed spatial analysis of IKT 16 for the first time. To provide further evidence of the extended emission, and to probe the source extent we created a radial profile centred on the point source from the data.\n We compared this with the two dimensional model of the point spread function (PSF) to look for an excess indicating the extended emission. Further, we tested for signatures of asymmetry along the two halves of the nebular emission. Finally we performed a morphological fitting of the elongated structure with a simple model of the nebula centred on the point source, and determined its geometrical parameters like the size, ellipticity and the rotation angle.\n\\subsection{Radial profile}\nRadial profiles were created up to 1$\\hbox{$^\\prime$~\\\/}$ in two energy bands (0.5--2 keV, and 2--7 keV) by extracting net counts in circular annuli centred on the point source using the tool { \\it dmextract} and then rebinning to reach a reasonable statistical precision. The background was extracted from a circular region of the same area, away from the source region but inside the SNR.\nThe PSF of the {observation~\\\/} was simulated\nusing the {\\it Chandra~\\\/} ray tracer chaRT \\footnote{http:\/\/cxc.harvard.edu\/chart\/} which simulates the High Resolution Mirror Assembly based on the energy spectrum of the source and the {observation~\\\/} exposure. The output of chaRT was modelled with the software MARX \\footnote{http:\/\/cxc.harvard.edu\/chart\/threads\/marx\/} taking into account the instrumental effects and the EDSER subpixel algorithm to be consistent with the observational data. The best-fit spectrum of the point source (see section \\ref{spec-pnt}) was used. Fig.~\\ref{radial} shows the radial profile of the {observation~\\\/} along with the simulated PSF. The data at both the energy bands are consistent with a point source up to a radius < 1\\hbox{$^{\\prime\\prime}$}~ beyond which it clearly has higher net counts than expected from a point source simulated at the same position and with the same spectral parameters. This corresponds\nto the nebular component. \nAlthough the geometrical model (see section \\ref{sec:spa-mod}) indicates that the source FWHM is about 5\\hbox{$^{\\prime\\prime}$}, this exercise indicates that the source extends further beyond. This excess is best seen in the energy range of 0.5--2 keV. Its brightness decreases outwards like $r^{-1.5}$. Figure~\\ref{images}d indicates that this further extension is probably elongated in the east-west direction like the main X-ray nebula and the radio nebula.\n \n\\subsection{Investigating signatures of asymmetry in the nebula morphology}\n\\label{east-west-counts}\nVisual inspection of the extended source near the centre of IKT 16 from Fig.~\\ref{images}a, gives an impression of an elongated symmetric structure. However, we looked for the possibility of an east-west asymmetry\nby dividing the nebular structure into two halves along the axis going through the point source, and compared the total counts from the two regions. The number of counts in the east and west nebula are $50 \\pm 9$ and $67 \\pm 10$ respectively indicating that the west nebula may be slightly brighter than its other half. It cannot be determined conclusively as the values are consistent within their errors. We investigated this issue further through spectral analysis of the same regions in section \\ref{spec-east-west}. \n\n\n\n\n\\begin{table}\n\\centering\n\\caption{Parameters of the best-fit model to {pulsar wind nebula~\\\/} in IKT 16.\nModel used is a 2-D Gaussian function for the nebula and a constant background.\nTheta is the angle between the major axis and the north direction.}\n\\begin{tabular}{c c c}\n\\hline\n\nParameter  & Value & Units\\\\\n \\hline         \nFWHM   & 5.2$^{-0.9}_{+1.0}$ & arcsec \\\\\nEllipticity & 0.6 $\\pm 0.1$ & \\\\\nTheta & 82 $\\pm 7$ & degrees \\\\\nAmplitude & 2.9 $\\pm 1 $ & counts\/pixel \\\\\nBackground & 0.12 $\\pm 0.01$ & counts\/pixel \\\\\n\\hline    \n\\end{tabular}\n\\\\\n\\label{tabsherpa}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45]{IKT16BkgSubtractedProfile.eps}\n\\caption{Radial counts profile of the source (blue star: 0.5--2 keV; red triangle: 2--7 keV) plotted against the 2-D PSF (0.5--8 keV, in black crosses), clearly showing the extended nature of the source.}\n\\label{radial}\n\\end{figure}\n\n\\subsection{Spatial modelling}\n\\label{sec:spa-mod}\nSpatial fitting was performed on a larger region of size 35\\hbox{$^{\\prime\\prime}$} $\\times$ 45\\hbox{$^{\\prime\\prime}$} centring the source to detect the existence of a possible diffuse\n emission underlying the nebular one, and also constrain the background better. The PSF described in the \nprevious section was loaded as a table model in {\\it Sherpa} to model the point source emission. The remaining excess consisted of the background and the emission from the nebula. This was modelled with a constant background and a 2-D unnormalized Gaussian function (gauss2d model in {\\it Sherpa}), convolved with the PSF. In the initial iterations it was found that the position of the 2-D Gaussian used to model the nebula was consistent with the point source. In order to constrain the geometrical parameters of the nebula like its FWHM along its major axis, ellipticity and rotation angle better, its position was fixed to the point source henceforth.\nThe best-fit parameters were determined by the C-statistic \\citep{cash1979} and errors were estimated at 90$\\%$ confidence level. The residuals do not show any systematic pattern indicating that this analysis is not sensitive to any substructures apart from the elongated structure of the nebula. Fig.~\\ref{images}b shows the model with the contours from the data smoothed with a Gaussian of $\\sigma$ = 3\\hbox{$^{\\prime\\prime}$} overlayed on it. This highlights the fact that the model is a good description of the data. It is worth mentioning that we also tried to fit a 2-D Lorentz model with a varying power-law (beta2d model in {\\it Sherpa}) instead of the 2-D Gaussian used to model the nebula. We noticed that by freezing the rotation angle and ellipticity, the power-law index $\\alpha$ tends towards the maximum limit of 10 which closely approximates the 2-D Gaussian model. The difference in C-statistic between beta2d and gauss2d is 15 in the Lorentzian limit ($\\alpha$=1 in beta2d), and 2 in the Gaussian limit ($\\alpha$=10 in beta2d). We therefore concluded that the data does not favor broad wings and a 2-D Gaussian is preferred for the nebular emission.\n\nThe best fit parameters are listed in Table \\ref{tabsherpa}. The estimated FWHM of 5.2\\hbox{$^{\\prime\\prime}$}~is in agreement with that measured from the image of the source. It is inclined at an angle of 82\\hbox{$^\\circ$}~ between the long axis and the north direction and is aligned with the radio nebula (see Fig.~\\ref{radio}). \nAn ellipticity of 0.6 indicates a major to minor axis ratio of 0.8. Another important result is that there is no signature of displacement between the point source and the nebula.\nThe best-fit position of the centre of the nebula obtained corresponds to R.A.(J2000)=00:58:16.824 and DEC.(J2000)=-72.18:05.32 with an error of 0.1\\hbox{$^{\\prime\\prime}$} and 0.2\\hbox{$^{\\prime\\prime}$} on the x and y positions respectively. This is consistent with the position of the point source obtained with {\\it celldetect }.\n\n\n\\section{X-ray Spectral analysis}\n\\label{sec-spec}\nWe have performed detailed spectral analysis of the point source and the nebular component of IKT 16. In addition we have also investigated the diffuse component extending to larger scales.\nThe main deciding factors for the spectral analysis are the regions used for the spectral extraction and the background modelling, taking into account the contribution\nof the point source in the nebular spectrum and vice versa. This is described in the subsequent subsections. The analysis was performed in the energy range of 0.5--7 keV. C-statistic was used for spectral fitting and errors were estimated at 90$\\%$ confidence interval. To account for the photoelectric absorption by the interstellar gas, two absorption components were used as in \\cite{owen2011}. The first, {\\it phabs} component was fixed at the\nGalactic value of $6\\times10^{20}{\\rm ~erg~s^{-1}}$ (\\citealt{dickey90}), and a free absorption ({\\it vphabs}) component was used to account for absorption inside the SMC. This second component has metal abundances fixed at 0.2 solar, as is typical in the SMC (\\citealt{russell92}). \n\\subsection{Point source spectrum} \n\\label{spec-pnt}\nFor the point source, a circular region of radius 1.5 pixels was extracted centred on the best-fit coordinates of the source. The extraction region is shown in Fig.\n\\ref{images}c. A circular annulus outside the nebular extraction region was used for the background spectrum. Apart from this the\nastrophysical background due to the nebular contribution was also considered. For this we calculated the fraction of nebular emission contributing in the point source extraction region (from the best-fit morphological model of the source), and accounted for it as an additional model component in the spectral fitting, with parameters fixed to the best fit values obtained in section \\ref{spec-pwn}.\n\nWe tried to fit the spectrum with several models including an absorbed power-law, an absorbed blackbody and a combination of both models. The absorbed blackbody model leads to an unacceptable spectral fit with a difference of 54 in C-statistic value between the two models with respect to the power-law model. The absorbed power-law model provides a good fit as illustrated on Fig.~\\ref{spec-1}a. The best fit parameters along with the flux are tabulated in Table \\ref{table-specfit}. The addition of a blackbody component was not required and did not improve the fit significantly. Also, the additional absorption column density inside the SMC could not be constrained well and was prone to large error bars as can be seen from Table \\ref{table-specfit}. Its value was however\nconsistent with that obtained from the previous analysis of the {\\it XMM-Newton~\\\/} {observations~\\\/} by \\cite{owen2011} which had smaller errors associated with it probably due to the larger number of counts. \n\\begin{figure*}\n\\centering\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{pt_src_without_resi.ps}}\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{total_neb_new_resi.ps}}\n \\caption{The figures show the ACIS-S spectra for the putative pulsar (a,{\\it left}) and the entire PWN (b,{\\it right}).\nThe solid lines correspond to the respective best-fit spectral models. The plots have been rebinned for visual clarity.}\n\\label{spec-1}\n\\end{figure*}\n\\subsection{Nebula spectrum}\n\\label{spec-pwn}\nFor the spectral extraction of the extended source, a box region was used with the region corresponding to the point source excised (see Fig.~\\ref{images}c).\nIts size was optimized from the morphological modelling of the nebula, and its centre was made coincident with the point source. The background was taken from the same annular region that was used as a background for the point source spectral extraction. For the additional astrophysical background contributed by the point source leakage in the nebular region,\nwe extracted a spectrum from the events from the PSF simulated by chaRT with spectral parameters similar to those of the point source, in the same region that was used for the nebular spectral\nextraction, and added it to the internal background spectrum as an additional background component. \n\nWe used an absorbed power-law as the spectral model for all the nebular fits described henceforth. In contrast to the point source spectrum, the spectrum of the nebula is very soft. Keeping the absorption inside the SMC free in these fits leads to strong correlation between the absorption column density and the power-law index $\\Gamma$. To avoid this it is better to constrain this parameter from the point source spectrum, and freeze it to this value for the nebular spectral fits. However, the better capability of the {\\it XMM~\\\/} {observations~\\\/} of IKT 16 to constrain the local absorption density, as discussed earlier, led us to fix this value to that obtained from the previous analysis of IKT 16 with {\\it XMM-Newton~\\\/}\\citep{owen2011} in all the nebular fits. The spectrum along with its best-fit model and residuals is shown in Fig.~\\ref{spec-1}b, and the best fit parameters are tabulated in Table \\ref{table-specfit}.\n\\subsection{Outer and Inner nebula spectra}\n\\label{spec-pwn-inner}\nIn order to look for changes in the spectral parameters, particularly a spectral steepening with radius in the nebula, we divided the total nebular region into two parts, by choosing an inner annular region centred on the point source, and an outside region excluding it. The extraction regions are shown in Fig.~\\ref{images}c. The Fig.~\\ref{spec-2}a and \\ref{spec-2}b show the inner and outer spectrum respectively with their best-fit models and residuals, and Table \\ref{table-specfit} their best-fit parameters. Although the outer nebula is hinted to be softer then its inner counterpart by the absence of counts above 5 keV, we detect no steepening of the spectral index $\\Gamma$ which would be indicative of synchrotron cooling. The $\\Gamma$ values are consistent within errors at 90 $\\%$ confidence. \n\\subsection{East-west nebula spectra}\n\\label{spec-east-west}\nIn Sect.~\\ref{east-west-counts} where we report the extracted total number of counts in the east and west nebular region, we find evidence that the west of the nebula may be slightly brighter although the values are consistent within errors. In continuation, we also extract spectra from the same regions following the same procedure as in Secs.   \\ref{spec-pwn} and \\ref{spec-pwn-inner}. Figures~\\ref{spec-2}c and Fig.~\\ref{spec-2}d show the east and west spectrum respectively with their best-fit models and residuals, and Table \\ref{table-specfit} their best-fit parameters. As indicated by the earlier exercise in Sect. \\ref{east-west-counts}, the spectrum from the west of the nebula shows slightly higher value of flux but it is comparable to that measured from the east nebula within errors at 90 $\\%$ confidence level. We do not detect any significant change in the spectral index $\\Gamma$. \n\\subsection{Diffuse emission}\n\\label{diff-neb}\nFrom Figs.~\\ref{images}d and \\ref{radial}, we see evidence that the source extends further beyond the region adopted to study the nebula spectrum. To investigate further, we extracted the spectrum of the diffuse component by choosing an annular region excluding the nebular extraction region (box region in Fig.~\\ref{images}c) and extending up to the size of the {\\it XMM-Newton~\\\/} point source extraction region of 20\\hbox{$^{\\prime\\prime}$} \\citep{owen2011}.\nUsing the same spectral model as that for the nebular spectrum, we find that the diffuse component has a flux and spectral index (tabulated in Table \\ref{table-specfit}) comparable to that of the main nebular component. We also tried to fit the spectrum with a Sedov model, since that emission might be associated with the SNR. We fixed the parameters to those obtained by \\cite{owen2011} (the data quality does not allow fitting anything else than the normalisation). The resulting C-statistic is higher by 9 than that of the power-law model. This does not favor the thermal model, but does not allow ruling it out either. However, no other part of the SNR is as bright, and this diffuse emission is centred on the main X-ray nebula. So we favor the interpretation as a further PWN component.\n\nIn addition, we also investigated whether this emission could be a halo due to the scattering from the foreground dust in the interstellar medium. \\cite{predel1995} using \\emph{ROSAT} observations, studied X-ray scattering halos around 25 point sources including the three bright sources in the LMC, LMC X-1, LMC X-2 and LMC X-3. The local absorbing density around IKT 16 is lower that in LMC X-1, but higher than in LMC X-2 and LMC X-3, and thus lies in the range covered in this study. The authors found that the relative intensity of the scattering halos of the LMC sources w.r.t the point source are $\\sim$ 1\\%. This is much weaker than what we find for the diffuse component which is $\\sim$ 25\\% (< 2 keV) of the putative pulsar emission. Moreover, \\emph{ROSAT} observed a flat profile for the halos extending to > 100\\hbox{$^{\\prime\\prime}$}. Within our extraction region of the diffuse emission (20\\hbox{$^{\\prime\\prime}$}), the fraction is expected to be even lower than 1\\%. Hence the dust scattering origin for this diffuse emission is very unlikely.\n\nFinally, we compared our results with the spectral model of the point source seen with {\\it XMM-Newton~\\\/} \\citep[Table 3 in][counting the spillover of the point source into the SNR extraction region, and the SNR emission in the point source region]{owen2011}. The total flux measured in the energy range of 0.5--8 keV from the {\\it Chandra~\\\/} observation is $21.3 \\times 10^{-14}$\n${\\rm ~erg~cm^{-2}~s^{-1}}$ \n(putative pulsar + nebula + diffuse emission; see Table \\ref{table-specfit} for the obtained values) comparable to $24.9 \\times 10^{-14}$ ${\\rm ~erg~cm^{-2}~s^{-1}}$ measured from the {\\it XMM-Newton~\\\/} spectral model. The unabsorbed luminosity in the same energy range (0.5--8 keV) measured  from {\\it Chandra~\\\/}~ is \\textbf{1.0} $\\times10^{35}$ ${\\rm ~erg~s^{-1}}$. In comparison, the {\\it XMM~\\\/} spectral model corresponds to an unabsorbed luminosity of 1.1 and 1.3 $\\times10^{35}$ ${\\rm ~erg~s^{-1}}$ in the 0.5--8 keV and 0.5--10 keV energy ranges and is in agreement with the {\\it Chandra~\\\/} results. It should be noted that, although the total unabsorbed luminosity (0.5--10 keV, SNR + point source) in \\cite{owen2011} is consistent with the spectral model, there seems to be an error in quoting the unabsorbed luminosity of the point source as 1.6 $\\times10^{35}$ ${\\rm ~erg~s^{-1}}$ in that paper.\n\n\n\\begin{figure*}[htp]\n\\centering\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{inner_new_resi.ps}}\\quad\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{outer_new_resi.ps}} \\\\\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{nebula_east_spec_new_resi.ps}} \\quad\n\\subfigure[]{\\includegraphics[height=0.45\\textwidth,angle=-90]{nebula_west_spec_new_resi.ps}} \\\\\n\\caption{Same as in Fig.~\\ref{spec-1} but for the inner nebula {\\it a}, outer nebula {\\it b}, east nebula {\\it c} and west nebula {\\it d}. Please note that in the outer nebula, i.e. Fig.~{\\it b}, there are no events above 5 keV.}\n\\label{spec-2}\n\\end{figure*}\n\\begin{table*}\n\\caption{Parameters of the best-fit spectral model to putative pulsar and the nebula. Errors are quoted at 90 $\\%$ confidence.\nThe {\\it Chandra~\\\/} energy range is 0.5--8 keV.}\n\\centering\n\\begin{tabular}{lcccccc}\n\\hline\n\nRegion  & SMC absorption &  Power-law photon Index  & F$_{\\rm X}$$^{a}$ &  F$_{\\rm X}$$^{a}$ &  L$_{\\rm X}$$^{b}$\\\\\n   & $10^{21}$cm$^{-2}$  &  $\\Gamma$  & 0.5--8 keV &  2--10 keV &  0.5--8 keV \\\\\n\\hline\nPutative pulsar  & 5.3$^{+3.9}_{-3.5}$ & 1.11$\\pm0.23$ & 16$\\pm 1$  & 17$\\pm2$ & 7.20\\\\\nNebula & 3.4 (f)$^{c}$  & 2.21$^{+0.40}_{-0.37}$ & 2.2$\\pm0.2$  & 1.3$^{+0.4}_{-0.5}$ & 1.19 \\\\\nInner nebula & 3.4 (f)$^{c}$ & 2.22$^{+0.48}_{-0.45}$ & 1.6$\\pm0.3$  & 1.0$^{+0.5}_{-0.7}$ & 0.78\\\\\nOuter nebula & 3.4 (f)$^{c}$ & 2.18$^{+0.60}_{-0.54}$ & 0.75$^{+0.38}_{-0.41}$ & 0.47$\\pm0.63$ & 0.41\\\\\nWest nebula & 3.4 (f)$^{c}$  & 1.97$^{+0.52}_{-0.40}$  & 1.4$\\pm 0.3$ & 0.9$^{+0.5}_{-0.6}$ & 0.67 \\\\\nEast nebula & 3.4 (f)$^{c}$  & 2.23$^{+0.67}_{-0.47}$ & 0.93$^{+0.4}_{-0.36} $  & 0.55$\\pm0.67$ & 0.45 \\\\\nDiffuse emission & 3.4 (f)$^{c}$ & 2.20$^{+0.38}_{-0.30}$ & 3.1$\\pm0.2$ & 1.9$^{+0.3}_{-0.2}$ & \\textbf{1.70} \\\\\n\\hline \n\\label{tabspec}   \n\\end{tabular}\n\\\\\n$^{a}$ - Observed flux in units of $10^{-14}$ ${\\rm ~erg~cm^{-2}~s^{-1}}$ \\\\\n$^{b}$ - absorption corrected luminosity in units of $10^{34}$ ${\\rm ~erg~s^{-1}}$ assuming a distance of \\textbf{61} kpc. \\\\\n(f) - Parameter fixed for consistency between fit regions. \\\\ \n$^{c}$ - SMC absorption for all nebular spectra was frozen to the column density obtained from the {\\it XMM-Newton~\\\/} {observation~\\\/}. \\\\\n\\label{table-specfit}\n\\end{table*}\n\\section{Radio}\nThe PWN in the radio-continuum regime extends far beyond X-ray detection as can be seen in Fig.~\\ref{radio}. The radio extent of the source as measured visually, is $70\\pm10$\\hbox{$^{\\prime\\prime}$}.\nWe note that the X-ray point source correlates to one of the peaks in our RC image. However, no corresponding point source can be drawn anywhere near the X-ray point source.\n\nWe point to the new high resolution data (Fig.~\\ref{radio}) where two lobe like features, somewhat symmetric, each $20\\pm5$\\hbox{$^{\\prime\\prime}$}~in extent, outside of the X-ray nebula are seen. These radio-continuum features oppose our initial scenario that the radio nebula has been pushed aside by the reverse shock \\citep{owen2011}. It seems it is more likely that we are still seeing the nebular expansion in the cold ejecta. \n\nAcross the 2.1\\,GHz band, the radio spectral index is flat as reported in \\cite{owen2011}. Somewhat surprising, we did not detect any polarisation in either {\\it Q}, {\\it U} or {\\it V} stokes parameters even that our detection level is better than 1\\%. The small scale structure may play an important role in smoothing out any weak polarisation. At the same time, depolarisation may reveal regions of underlying turbulence and\/or compression and heating of thermal material at various shocks within the remnant system \\citep{anderson1995}. This depolarisation may also indicate somewhat older age of the remnant. The noticeably younger PWNe in the LMC such as N157B \\citep{lazendic}, SNR J0453-6829 \\citep{hab2012}, and 0540-69.3 \\citep{bran2014} all have strong polarisation. \n\\section{Discussion}\n\\label{sec:disc}\nConfirmation of the PWN nature of the source makes IKT 16 the first composite SNR to be discovered in the SMC.\nThe nebular structure is consistent with the presence of an elongated PWN around a putative pulsar, which extends further out in the form of a fainter diffuse emission.\nHowever, we cannot rule out the presence of an inner compact X-ray nebula of the order of 0.1 pc which cannot be resolved at the distance of the SMC. \nTherefore, the overall impression is that of the bright central point source corresponding to a putative pulsar or a putative pulsar + unresolved inner PWN component, and an  extended emission component representing the PWN.\n\n Although the present {\\it Chandra~\\\/} {observation~\\\/} in full-frame mode is not capable of detecting pulsations from the central putative pulsar, we can nevertheless infer its properties to a certain extent by assuming some basic characteristics of the neutron star as is described below. In the later subsections, we also discuss the evolutionary stage of the {pulsar wind nebula~\\\/} as expected from its morphology.\n\\subsection{Derived properties of the pulsar}\n\\label{sec:discsnr}\n\nThe nebular emission in X-rays reflects the youngest generation of the emitting particles. Therefore, a strong correlation is expected \nbetween the non-thermal X-ray luminosity of the PWN and its pulsar, with the spin down properties of the\npulsar itself. The properties of the putative pulsar in IKT 16 were estimated adopting the work of \\cite{pwnchandra}. In this paper the correlation between the luminosity of the PWN, the non-thermal luminosity of the pulsar, and its various properties are derived from a large sample of {\\it Chandra~\\\/} {observations~\\\/} of PWNe. The unabsorbed luminosities of the PWN and the pulsar in IKT 16 are reported in Table \\ref{tabspec}. Using these values and the tabulated properties of the PWNe from \\cite{pwnchandra}, we estimated the spin-down power of the putative pulsar by comparing the total non-thermal luminosity of the sources (PWN+pulsar component) with their spin-down power. This ensured our estimate was free from the assumption that the central source is exclusively the pulsar and does not contain a compact nebular component. It is noteworthy to mention that in order to perform this estimation, we have not accounted for the flux from the diffuse emission (described in section \\ref{diff-neb}) which is not as well measured and whose origin is less unambiguous. At the same time addition of this component would not alter our results in the limits of the correlation measured in \\cite{pwnchandra}. Fig.~\\ref{corr} plots the spin-down power against the total luminosity for all the objects from \\cite{pwnchandra}, with a line drawn to indicate the total luminosity of the central source in IKT 16. It corresponds to an expected spin-down power $\\dot{E} \\sim 10^{37} {\\rm ~erg~s^{-1}}$.  Further, assuming its age to be the same as that measured from the SNR \\citep{owen2011}, ${\\it i.e.~\\\/}$ 14.7 kyr, we overlayed the central source on the $P - \\dot{P}$ diagram from the ATNF catalogue \\citep{atnf}. The expected spin period is < 100 ms with $\\dot{P} \\sim 10^{-13}$ s s$^{-1}$. This points towards a young and energetic pulsar powering the PWN in IKT 16, in fact the youngest in the SMC. Future on-axis {\\it XMM~\\\/} {observations~\\\/} or {\\it Chandra~\\\/} {observations~\\\/} using operating modes with better timing resolution can be used to search for pulsations from this source.\n\\begin{figure}\n\\centering \n\\includegraphics[height=0.45\\textwidth,angle=-90]{correlation-psr-edot-new.ps} \n\\caption{ Dependence of the total non-thermal luminosity (nebula+pulsar) on the pulsar spin-down power for the PWNe observed with {\\it Chandra~\\\/}. Adapted from \\cite{pwnchandra}. The red dashed line indicates the luminosity of the putative pulsar + PWN in IKT 16.}\n\\label{corr}\n\\end{figure}\n\\subsection{Energy spectrum}\n\\label{discspec}\nThe spectra of both the central and the extended source can be satisfactorily fitted with a power-law model, expected from a synchrotron emission dominated spectrum. The hard power-law spectral index 1.1 for the point source resembles that of a typical young rotation powered pulsar where the non-thermal radiation is generated by the particles accelerated in the pulsar magnetosphere \\citep[][and references therein]{michel1992}.\nNo additional thermal component (which would denote the emission from the surface or the polar caps of the neutron star) was required to fit the pulsar spectrum. The point source is  also three times brighter than the surrounding nebula. If this comprises exclusively the pulsar emission, it is rather unusual since the average value of the nebular to pulsar emission is $\\langle\\frac{\\eta_{pwn}}{\\eta_{psr}}\\rangle \\sim 4 $ for a large sample of PWNe observed with {\\it Chandra~\\\/} \\citep{pwnchandra}. The $\\Gamma$ of the nebular emission is compatible with that usually observed for PWNe \\citep[1.5 to 2.1:][]{pwnchandra,li2008}, and is also consistent with particle wind models of Fermi shock acceleration \\citep{2001MNRAS.328..393A}.\n\\subsection{Morphology}\n\\label{sec:discsnrmorp}\nThe morphology of the PWN is elongated along the east-west direction as reported in section \\ref{spec-east-west}. \nThe obtained size of  5.2\\hbox{$^{\\prime\\prime}$}~of the X-ray nebula corresponds to a source extent of 1.5 pc at the distance of the SMC. The radio extent of the source is much larger corresponding to 21 pc with the brightest emission from the radio lobes, each $\\sim$ 6 pc in extent assuming the same distance.\nThis scenario is consistent with a longer cooling time for radio emission compared to that of X-ray emission.\nIn addition, the radio nebula extends even further than the estimated SNR centre, consistent with the fact that the earliest electrons were produced when the pulsar was much closer to the SNR centre. Further, the long axis of the radio and X-ray nebula are aligned with each other. This axis of alignment points to the centre of the SNR, so is presumably aligned with the pulsar's direction of motion.\n\\subsection{Interaction of the SNR with the PWN  and its stage of evolution}\n\\label{pwn-interact}\nThe SNR in IKT 16 is in the adiabatic Sedov phase of evolution with an age of $\\sim$ 14.7 kyr \\citep{owen2011}. The shell of the SNR has important implications on the PWN, and the physics of these systems is extremely complicated due to the rapid evolution of both the SNR and the central pulsar \\citep{gaensler2006,gelfand2007,gelfand2009}. The evolution of a PWN inside an SNR follows three important evolutionary phases: an initial free-expansion in the supernova ejecta, the collision between the PWN and SNR reverse shock, which crushes the PWN subjecting it to various instabilities, and eventually subsonic re-expansion of the PWN in the shock heated ejecta \\citep[][and references therein]{2001ApJ...563..806B,2003A&A...397..913V,2003A&A...404..939V,2004A&A...420..937V,gelfand2007}. After the reverse shock interaction has died off, ${\\it i.e.~\\\/}$ in the PWN re-expansion stage, the neutron star is often displaced from its PWN, leaving behind the \"relic\" PWN usually observed in radio and forms a \"new\" PWN comprising of freshly injected particles, and therefore observable in the X-rays. A later additional phase of \"bow-shock\" nebula can sometimes be identified, as the local sound velocity decreases with the pulsar's progression through the SNR. In this phase the pulsar's motion may become supersonic when it approaches the shell of the SNR, and it may acquire a bow-shock morphology \\citep{2003A&A...397..913V,2003A&A...404..939V,2004A&A...420..937V}.\n\nAdopting the total mechanical energy released in the explosion ($E_{51}$ in units of $10^{51}$ erg) and the ambient medium density ($n_{0}$) from the adiabatic Sedov modeling in \\cite{owen2011}, and assuming the mass of the ejecta, $M_{ej} = 10 M_{\\odot}$, we can calculate the reverse shock trajectory \\citep{mckee1995} at the present age of the SNR (14.7 kyr). At the distance of the SMC, we obtain that the reverse shock radius $R_{s}$ is 11 pc away from the centre of the SNR. This would imply that although it might not have encountered the smaller X-ray nebula yet \\citep[measured to be at 8 pc from the centre of the SNR][]{owen2011}, it would have crossed the larger radio extent of the nebula and interacted with the lobes. This picture is however not supported by the new high resolution radio data. The presence of two bright radio lobes more or less symmetric about the X-ray counterpart is consistent with an expansion of the PWN into cold ejecta in homologous expansion. \nWe however point out that there may be uncertainties in measurement of our Sedov model parameters for the SNR IKT 16 \\citep{owen2011} which may limit the estimation of $R_{s}$.\n\\section{Conclusions}\n\\label{sec:conc}\nThanks to the high resolution {\\it Chandra~\\\/} ACIS {observation~\\\/} we have been able to resolve the hard X-ray emission near the centre of the SNR IKT 16 into a PWN centred on a\nputative pulsar. We have imaged the PWN, constrained its geometrical parameters, and have measured all the  spectral components separately. With new high resolution radio data we have determined the radio morphology and its extent precisely. The main results can be summarized as follows:\n\\begin{itemize}\n\\item The putative PWN is elongated but centred on the point source. Comparison with the radio counterpart indicates that the reverse shock may not have yet met with the PWN surface.\n\\item The point source at the centre is about three times brighter than the elongated feature.\n\\item Morphological modelling of the X-ray nebula with a PSF and a 2D Gaussian resulted in an FWHM of 5.2\\hbox{$^{\\prime\\prime}$}~for the PWN with its axis aligned with the larger radio nebula.\n\\item The point source at the centre has a much harder spectrum than the extended emission from the nebula. This points to the presence of a \npulsar dominated by non-thermal emission. The expected $\\dot{E}$ is $\\sim 10^{37} {\\rm ~erg~s^{-1}}$ and spin period < 100 ms, with  \n$\\dot{P} \\sim 10^{-13}$ s s$^{-1}$. This points towards a young and energetic pulsar powering the PWN in IKT 16.\nHowever, the presence of a compact X-ray PWN unresolved by Chandra at the distance of the SMC cannot be ruled out.\n\\end{itemize} \n\n\n\n\n\n\n\\begin{acknowledgements}\nThis research has made use of data obtained from the {\\it Chandra~\\\/} X-ray observatory. We used the KARMA and MIRIAD software packages developed by the\nATNF. The Australia Telescope Compact Array is part of the Australia\nTelescope which is funded by the Commonwealth of Australia\nfor operation as a National Facility managed by CSIRO. CM acknowledges Fabio Acero for very useful discussions and comments on the paper. The authors would like to acknowledge the referee Satoru Katsuda for very useful comments which significantly improved the paper.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\bigskip\n\n\\subsection{The problem. Results and conclusions}\n\nThe clue issue in checking quantum electrodynamics (QED) is the measurement\nof the magnetic moment of electron with the subsequent comparison of its\nmeasured value with the anomalous electron magnetic moment calculated via\nthe Standard Model that is mostly QED in this case. Up to now, within every\nexperimental and theoretical accuracy achieved, these two values do\ncoincide. The allowed, within the errors, discrepancy between the\nexperimental and theoretical values of the electron magnetic moment is\nexpected to be diminishing on and on, as the precision grows, and hopefully\nthe coincidence between them will be maintained with better and better\naccuracy. On the other hand, as far as one is seeking for possible\ntheoretical amendments to the Standard Model, admissible within the above\nsituation, one should confine their impact on the electron magnetic moment\nto lie within the present experimental and theoretical indeterminacy. A\ncertain candidate for going beyond the standard QED is proposed by the\nnoncommutative (NC) electrodynamics. It was found recently \\cite{AGSV} that\nin the framework of that theory a static classical charge at rest \\ carries\na magnetic moment, called NC magnetic moment, whose smallness is determined\nby a noncommutativity parameter $\\theta,$ supplying the theory with the\nfundamental length \\footnote\nThe noncommutativity by no means is the only method of introducing the\nfundamental, or elementary, length into a theory. Throughout this paper,\nhowever, we shall mean namely NC fundamental length when using this notion.\nOn the other hand, we do not know whether the fundamental length as it is\nproposed by the noncommutativity mechanism is universal for all particles\nand fields. For this reason we shall discuss its values independently when\nwe deal with different particles.} $l=\\sqrt{\\theta}$. By demanding that, for\nthe electron, the NC magnetic moment be less than the existing error in\nmeasuring the electron magnetic moment we get an estimate from above on the\nparameter $\\theta$ and the associated fundamental length $l$ . Certain\nrestrictions on the fundamental length inherent in the NC theory also follow\nfrom the existence of the NC magnetic moment of heavier charged particles.\nHowever, the consideration of the noncommutative magnetic moment of the\nproton and of its contribution to the hyperfine splitting of the energy\nlevel $1S_{1\/2}$ in a hydrogen atom did not lead \\cite{AGSV} to any\nessentially new estimate for the maximum fundamental length. On the\ncontrary, consideration of leptons did.\n\nOnce the NC magnetic moment is found to be inversely proportional to the\nsize of the electric charge, an important role in getting this estimate is\nplayed by the size attributed to an electron, the smaller the size, the\nlarger the NC magnetic moment, the smaller the upper estimate on the NC\nparameter and the fundamental length. We probe different assumptions\nconcerning the \\textquotedblleft electron size\\textquotedblright , the\nultimate one being that it is restricted from below only by the fundamental\nlength $l$ itself, since neither object is supposed to be smaller than it.\nIn this way a hitherto lowest upper bound on the fundamental length, as it\nappears in the noncommutative field theory, was achieved in \\cite{AGSV}. On\nthe other hand, after we update the famous electron size estimate\\ due to\nBrodsky-Drell-Dehlmet \\cite{BrodDrell}, \\cite{Dehmelt} (not based on any\nnoncommutative mechanism, but only on a consideration of a possible\ncompositeness, or divisibility of the electron) by taking into account the\nmost recent measurements of the electron magnetic moment, we find the\nelectron size results to be two orders of magnitude smaller than the boldest\nestimate of the fundamental length obtained from speculations on\nnoncommutative magnetic moment. As far as in an NC field theory no size of\nany physical object is admitted to be smaller than the fundamental length,\nthis means that no more than 1\/100 part of the existing indeterminacy in the\nknowledge of the electron magnetic moment may be at the best ascribed to the\ncontribution of the noncomutative magnetic moment. Then, two options remain.\nEither there should be an extra extension beyond the standard QED, other\nthan NC electrodynamics, that may take responsibility for the main part of\nthe admitted, if any, deviation of the magnetic moment from the QED result,\nor, what is more probable, this admitted deviation will be essentially\nreduced by further more precise measurements.\n\nThe same analysis is repeated in the paper as applied to the $\\mu $-meson.\nThe crucial difference with the electron case is that the difference between\nthe theoretical and experimental values of the muon magnetic moment exceeds\nthe limits admitted by the errors. So, no further technical advancement is\nexpected to be able to remove this contradiction, and our results make us\nmore definite in claming that the noncommutativity cannot provide for the\nmissing part of the muon magnetic moment, a different way for extending the\nStandard Model remaining to be needed.\n\n\\bigskip\n\n\\subsection{Noncommutative magnetic moment}\n\nIn \\cite{AGSV}, classical field equations in $U(1)_{\\star}$-theory\n(noncommutative electrodynamics) were formulated that -- at least within the\nfirst order in the noncommutativity parameter $\\theta$ -- restrain the gauge\ninvariance in spite of the presence of external current, known to violate it\n(at least off-shell). By solving these equations electromagnetic field\nproduced by a finite-size static electric charge was found, and the fact\nthat this charge possesses a magnetic moment depending on its size was\nestablished. Let the external current in the field equations of NC\nelectrodynamics be just a static electric charge distributed inside a sphere\nof a finite radius $a$ with the uniform charge density \n\\begin{equation*}\n\\rho\\left( \\mathbf{r}\\right) =\\frac{3}{4\\pi}\\frac{Ze}{a^{3}}\\,,\\ \\ r<a,\\quad\nr=|\\mathbf{r}|\\,.\n\\end{equation*}\nOutside the sphere there is no charge: $\\rho\\left( \\mathbf{r}\\right) =0,$ if \n$r>a$. The above finite-size static total charge $Ze,$ where $e$ is the\nfundamental charge, produces not only the electrostatic field, but also\nbehaves itself as a magnetic dipole with the magnetic field given in the\nremote region $r\\gg a$ by the following vector-potentia\n\\begin{equation}\n\\hspace{0cm}\\mathbf{A}=\\frac{\\left[ \\mathbf{M}\\times\\mathbf{r}\\right] }{r^{3\n},\\ \\ \\mathbf{M}=\\boldsymbol{\\theta}(Ze)^{2}\\frac{2e}{5a}\\,,\n\\label{magnmoment}\n\\end{equation}\nwhere $\\mathbf{M}$ was called NC magnetic moment of the charged particle.\nHere the three spacial components of the vector $\\boldsymbol{\\theta}$ are\ndefined as $\\theta^{i}\\equiv(1\/2)\\varepsilon^{ijk}\\theta^{jk},$ $i,j,k=1,2,3$\nin terms of the antisymmetric noncommutativity tensor $\\theta^{\\mu\\nu}$ that\nfixes the commutation relations between the operator-valued coordinate\ncomponents $[X^{\\mu},X^{\\nu}]=i\\theta^{\\mu\\nu},$ and only the space-space\nnoncommutativity, i.e. the special case of $\\theta^{0\\nu}=0$ in a certain\nLorentz frame, was considered.\n\nThe extension (size) $a$ of the charge in (\\ref{magnmoment}) should be kept\nnonzero in the spirit of NC theory that does not admit objects with their\nsize smaller than the fundamental length $l=\\sqrt{\\theta}$, where $\\theta =\n\\boldsymbol{\\theta}|$. For a point charge a magnetic solution also exists \n\\cite{Stern}, although in this case it is not a magnetic dipole one. What is\nmore important is that that solution is too singular in the point $r=0$,\nwhere the charge is located, and hence it cannot be given a mathematical\nsense in terms of the distribution theory in a conventional way.\n\nIf we understand the radius $a$ in (\\ref{magnmoment}) as the size of an\nelectrically charged fundamental particle ($Z=1$), we can speculate on what\nthe contribution of the noncommutativity into its magnetic moment $\\mathbf{M}\n$ may be. Certainly, this is expected to be very small, because of the\nextreme smallness of the noncommutativity parameter $\\theta $. It is\nprimarily supposed \\cite{DFR} that the corresponding length $l=\\sqrt{\\theta }\n$ should be of the Plank scale of $l\\sim 10^{-33}$ cm (or $\\Lambda _{\\mathrm\nPl}}\\sim 4\\cdot 10^{19}$ Gev in energy units). The reason is that at so\nsmall distances unification of gravity with quantum mechanics requires\nquantization of space-time. Although the Plank scale is far beyond any\nexperimental reach, the everlasting problem is to estimate the upper limits\non $\\theta $ basing on the existing and advancing experimental preciseness.\nIn \\cite{AGSV} it was discussed what new restrictions on the extent of\nnoncommutativity may follow from the newly established fact that a charged\nfundamental particle is a carrier of the magnetic moment (\\ref{magnmoment})\nin an NC theory, irrespective of its orbital momentum or spin. In the\npresent article we shall further elaborate this matter addressing the\ncharged leptons $e$ and $\\mu $ as the \\textquotedblleft\nsmallest\\textquotedblright\\ -- and hence providing the maximum contribution\nof (\\ref{magnmoment}) -- particles, to leave alone quarks -- also small, but\nwhose magnetic moment is beyond measurements.\n\n\\section{Upper bounds for fundamental length from noncommutative magnetic\nmoment}\n\n\\subsection{Limitations based on high-energy scattering estimates of lepton\nsizes}\n\nIn high-energy electron-positron collisions leptons manifest themselves as\nstructureless particles (see e.g. \\cite{BrodDrell} for an early discussion\nof this point), described by a fundamental (local), not composite field. No\ndeviation from this rule has been up to now reported. Taking the LEP scale\nof 200 Gev as an upper limit, to which this statement may be thought of as\nchecked, we must accept that a possible structureness of these leptons is\nbelow the length (call it divisibility length) $r_{0}=10^{-3}Fm$. In our\nfurther consideration we identify the charge extension $a$ with the\ndivisibility length, because it is hard to imagine a region occupied by a\ncharge that extends above this length, but cannot be divided into parts. (If\nit could, either the resulting charge would acquire a continuous value,\nsmaller than $e$, which contradicts basic assumptions, or the resulting\ncharge would occupy a smaller space and we would be left again with smaller \na$, down to the divisibility length.)\n\n\\subsubsection{Electron}\n\nBearing in mind that, for electron, the existing local theory perfectly\nexplains the value of its magnetic moment $M_{\\mathrm{e}}$, we expect that\nthe noncommutativity might only contribute into the experimental and\ntheoretical uncertainty $\\delta M_{\\mathrm{e}}$ existing in measuring and\ncalculating this quantity. A recent direct measurement of the anomalous\nmagnetic moment of electron, using the magnetic resonance spectroscopy of an\nindividual electron in the Penning trap \\cite{Dehmelt}, gives the result \n\\cite{Hanneke}, \\cite{nakamura\n\\begin{equation}\n\\left. \\left( \\frac{M_{\\mathrm{e}}}{\\mu }-1\\right) \\right\\vert _{\\mathrm{MRS\n}=0.00115965218073\\pm 28\\cdot 10^{-14},  \\label{hanneke}\n\\end{equation\nwhere $\\mu =e\/2m$ is the Bohr magneton. On the other hand, a new report \\cit\n{Bouchendira} appeared on an \\textit{independent} experimental determination\nof the same magnetic moment with a matching accuracy, obtained with the use\nof a measurement of the ratio $h\/m_{\\mathrm{Rb}}$ between the Plank constant\nand the mass of the $^{87}$Rb atom. The result is \n\\begin{equation}\n\\left( \\left. \\frac{M_{\\mathrm{e}}}{\\mu }-1\\right) \\right\\vert _{\\mathrm{Rb\n}=0.00115965218113\\pm 84\\cdot 10^{-14}.  \\label{theor}\n\\end{equation\nAuthors of \\cite{Bouchendira} fit the value of the fine structure constant \n\\alpha $ in such a way as to make (\\ref{theor}) coincide with the\ntheoretical prediction for the electron anomalous magnetic moment,\ncalculated (see \\cite{Mohr} for a review) with the accuracy, including QED\ncalculations up to $(\\alpha \/\\pi )^{4}$, also electroweak and hadronic\ncontributions (this fit leads to the so far most precise value $\\alpha\n^{-1}=137.035999037(91)$). For this reason the value (\\ref{theor}) is\nreferred to as \\textquotedblleft theoretical\\textquotedblright . (Certainly,\nthe roles of (\\ref{theor}) and (\\ref{hanneke}) might be reversed.) The\ntheoretical, (\\ref{theor}), and experimental, (\\ref{hanneke}), values of the\nelectron magnetic moment do not contradict each other, demonstrating the\nhitherto best confirmation of QED. The discrepancy between the\n\\begin{equation}\n\\frac{\\delta M_{\\mathrm{e}}}{\\mu }\\sim 10^{-12}  \\label{delta}\n\\end{equation\nlies within the accuracy of measurements and calculations. We demand that a\npossible contribution of the noncommutative magnetic moment in (\\re\n{magnmoment}) should not exceed it\n\\begin{equation}\n\\frac{\\delta M_{\\mathrm{e}}}{\\mu }>\\alpha \\theta \\frac{4m}{5a}\\,,\\ \\alpha\n=e^{2}\\,.  \\label{delM}\n\\end{equation\nWith the high-energy restriction on the size $a<r_{0}$ accepted above, Eq.\n\\ref{delM}) implies $\\theta <\\frac{\\delta M_{\\mathrm{e}}}{\\mu }(5r_{\\mathrm{\n}}\/4m\\alpha )$. As $r_{0}\\sim 10^{-3}Fm,$ we get from here and from\\ (\\re\n{delta})\\ the restriction on the fundamental length $l=\\sqrt{\\theta }<7\\cdot\n10^{-6}Fm=(28~Tev)^{-1}.$\n\n\\subsubsection{Muon}\n\nThe matters stand differently with the $\\mu $-meson. In the literature, its\nanomalous magnetic moment is calculated via the Standard Model with the\ninclusion of the QED lowest-order $\\mu $-$\\gamma $ vertex, $Z$-boson,\nneutrino and hadron lines. The deviation of the measured magnetic moment $\\\nM_{\\mu \\text{ }}$from the result of calculations makes the value (see A.\nHocker's and W.J. Marciano's 2009 update in \\cite{nakamura}, also \\cite{Mohr}\nfor a later detailed account) \n\\begin{equation}\n\\frac{\\delta M_{\\mu }}{\\mu }\\simeq 25\\cdot 10^{-10}.  \\label{deltamu}\n\\end{equation\nThis exceeds about 3.2 times the estimated 1$\\sigma $ error \\cite{nakamura}.\nIt is believed that this discrepancy may be overcome by including\nsupersymmetry for amending the theoretical result. If, on the contrary, we\ntry to explain this discrepancy by the effect of NC magnetic moment of the\nmuon, we get in the way similar to the one described above in this\nSubsection, using (\\ref{deltamu}) and the same indevisability length \nr_{0}\\sim 10^{-3}Fm$ that $l$ is smaller than $2.8\\cdot 10^{-5}Fm=$ (7 $Tev$\n$^{-1}$ as the high-energy based estimate.\n\n\\subsection{Ultimate estimates}\n\nOnce there is no evidence for any electron extension, it is worth admitting\nthat it may be only restricted by the fundamental length. Then, using $a=l\n\\sqrt{\\theta}$ in (\\ref{delM}) and the indeterminacy (\\ref{delta}), we\nobtain the ultimate bound of $l<6.6\\cdot10^{-8}Fm=(3\\cdot10^{3}$ $Tev)^{-1}\n. Dealing with the muon in the same way, but referring to (\\ref{deltamu})\ninstead of (\\ref{delta}), we obtain the ultimate estimate of $8\\cdot\n10^{-7}Fm=(240$ $Tev)^{-1}$.\n\n\\section{Upper bounds on fundamental length versus compositeness sizes of\nleptons}\n\nThere are \\cite{BrodDrell} much stronger restrictions on the lepton sizes\nthan those following from the high-energy collision experiments. These\nextend to the energy scale far exceeding the accelerator means. The point is\nthat if one imagines a lepton as a bound state of much heavier particles so\nthat the binding energy compensates the most part of their masses to make\nthe resulting state light, the Bohr radius $R$ of the composite state -- to\nbe treated as its size -- would be much smaller than the Compton length of\nthe lepton $\\lambda_{\\mathrm{C}}$. According to the Drell-Hearn-Gerasimov\nsum rule (see \\cite{BrodDrell} for references) the deviation of the\nanomalous magnetic moment $(M\/\\mu-1)$ from its QED value is proportional to\nthe ratio $R\/\\lambda_{\\mathrm{C}},$which is the measure\\ of compositeness.\nBased on the experimental data on magnetic moments of the known composite\nparticles -- proton and triton -- plotted against their measured sizes, a\nconjecture was formulated by Dehmelt \\cite{Dehmelt} that the proportionality\ncoefficient should be of the order of unity. Then, $R=\\lambda_{\\mathrm{C\n}\\delta M\/\\mu$.\n\n\\subsection{Electron}\n\nReferring to Eqs. (\\ref{hanneke}, \\ref{theor}) and using (\\ref{delta}) we\nmay update Dehmelt's 1988 result for the electron of $R<4\\cdot10^{-8}Fm$ to \nR<4\\cdot10^{-10}Fm$. This is two orders of magnitude smaller than our\nultimate estimate of $6.6\\times10^{-8}Fm$ for the fundamental length\nobtained in Subsection \\textit{2.2}. (The use of the assertion $R=\\lambda _\n\\mathrm{C}}\\delta M\/\\mu$ together with (\\ref{delM}) would result in the\ncondition $l<\\sqrt{5\/8\\alpha}(\\delta M\/\\mu)\\lambda_{\\mathrm{C}}=9.25R$,\nweaker than the already accepted condition that the fundamental length \\\nshould be smaller than any size, including the composite electron radius $R\n, that is to $l<R$ $.$) So, Dehmelt's conjecture provides a stronger bound\non the fundamental length, than the noncommutative magnetic moment. This\nmeans that no more than $10^{-2}$ part of the measured difference (\\ref{delM\n) may be at the most attributed to noncommutative contribution.\n\n\\subsection{Muon}\n\nThe muon radius estimated analogously, basing on the compositeness arguments\nand on the theory-experiment discrepancy (\\ref{deltamu}), gives the result \nR_{\\mu }\\simeq 0.5\\cdot 10^{-8}Fm.$ This is smaller than the ultimate\nestimate of Subsection 2.2 based on muon data. Again, once the muon size\ncannot be smaller than the fundamental length, this result indicates that\nthe NC magnetic moment alone definitely cannot take on the responsibility\nfor the discrepancy (\\ref{deltamu}) between the theory and experiment, and\nhence deviations from the Standard Model other than NC electrodynamics are\nneeded. Unlike the electron case above, one cannot set one's hopes upon the\nfuture growth of precession of measurements to abandon this conclusion.\n\n\\section*{Acknowledgements}\n\nT. Adorno thanks FAPESP, D. Gitman thanks CNPq and FAPESP for permanent\nsupport, A. Shabad thanks FAPESP for support and USP for the kind\nhospitality extended to him during the fulfillment of this work. A. Shabad\nalso acknowledges the support by RFBR under the Project 11-02-00685-a.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}