diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfgqo" "b/data_all_eng_slimpj/shuffled/split2/finalzzfgqo" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfgqo" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\n\n\\subsection{Generalized diagonals for rational billiards}\\label{subsec:gen:diag} Let $P$ be a Euclidean polygon with angles in $\\pi\\bb Q$. We call such a polygon \\emph{rational}. A classical dynamical system is given by the idealized motion of a billiard ball on $P$: the (frictionless) motion of a point mass at unit speed with elastic collisions with the sides. \n\nA \\emph{generalized diagonal} for the polygon $P$ is a trajectory for the billiard flow that starts at one vertex of $P$ and ends at another vertex. Since the group $\\Delta_P$ generated by reflections in the sides of $P$ is finite, the \\emph{angle} of a trajectory is well defined in $S^1 \\cong S^1\/\\Delta_P$. A motivating question for our paper is the following: \\emph{how close in angle can two generalized diagonals of (less than) a given length be} (in terms of the length)?\n\nMasur~\\cite{Masur} showed that the number of generalized diagonals of length at most $R$ grows quadratically in $R$. We will show, for some families of billiards, that the smallest gap $\\gamma^P_R$ between two generalized diagonals on $P$ of length at most $R$ satisfies \\begin{equation}\\label{eq:billiards:smallgap} \\lim_{R \\rightarrow \\infty}R^2\\gamma^P_R =0,\\end{equation} and for other specific billiard tables that\\begin{equation}\\label{eq:billiards:nogap}\\liminf_{R \\rightarrow \\infty} R^2 \\gamma^P_R >0.\\end{equation}\n\n\\subsection{Organization of the paper}\\label{subsec:organization} In the rest of \\S\\ref{sec:intro}, we describe the moduli space $\\Omega_g$ of holomorphic differentials, and state the general versions of (\\ref{eq:billiards:smallgap}) and (\\ref{eq:billiards:nogap}). In \\S\\ref{sec:sl2:strata}, we recall the definition and properties of the $SL(2, \\bb R)$-action on $\\Omega_g$ and the decomposition of $\\Omega_g$ into strata, and discuss the connection to billiards in \\S\\ref{subsubsec:billiards:translation}. We state our main technical results and important applications in \\S\\ref{sec:results}. We prove these results in \\S\\ref{sec:axiom}, following the approach in~\\cite{EskinMasur}. In \\S\\ref{sec:measure:bounds}, we calculate bounds for measures of certain sets in $\\mathcal H$ and give examples of flat surfaces $\\omega \\in \\mathcal H$ with different types of gap behavior. We study a family of flat surfaces arising from rational billiards in \\S\\ref{sec:billiards}. \n\n\\subsection{Translation surfaces}\\label{subsec:holo} Let $\\Sigma_g$ be a compact surface of genus $g \\geq 2$. Let $\\Omega_g$ be the moduli space of holomorphic differentials on $\\Sigma_g$. That is, a point $\\omega \\in\\Omega_g$ is a equivalence class of pairs $(M, \\omega)$, where $M$ is a genus $g$ Riemann surface, and $\\omega$ is a holomorphic differential on $M$, i.e., a tensor with the form $f(z)dz$ in local coordinates, such that $\\frac{i}{2}\\int_{\\Sigma_g} \\omega \\wedge \\bar{\\omega} = 1$. \n\nTwo pairs $(M_1, \\omega_1)$ and $(M_2, \\omega_2)$ are equivalent if there is a biholomorphism $f:M_1 \\rightarrow M_2$ such that $f_{*} \\omega_1 = \\omega_2$. For notational purposes, we will simply refer to the pair $(M, \\omega)$ by $\\omega$. Given $\\omega \\in \\Omega_g$, one obtains (via integration of the form) an atlas of charts on $\\Sigma_g$ (away from the finite set of zeros of $\\omega$) to $\\bb C \\cong \\bb R^2$, with transition maps of the form $z \\mapsto z + c$. For this reason, we will refer to $\\omega \\in \\Omega_g$ as a \\emph{translation surface} of genus $g$.\n\nThese charts determine a unique flat metric on $M$ with conical singularities at the zeros of the differential $\\omega$. Geometrically, a zero of the form $z^k (dz)$ corresponds to a cone angle of order $(2k+2)\\pi$. Zeroes of $\\omega$ are \\emph{singular} points for the flat metric. We refer to non-singular points as \\emph{regular points}. The space $\\Omega_g$ can be decomposed naturally into \\emph{strata} $\\mathcal H$ (see \\S\\ref{sec:sl2:strata} for details), each carrying a natural measure $\\mu_{\\mathcal H}$. \n\n\\subsection{Saddle connections and cylinders}\\label{subsec:sc} Fix $\\omega \\in \\Omega_g$. A \\emph{saddle connection} on $\\omega$ is a geodesic segment in the flat metric connecting two singular points (that is, zeros of $\\omega$) with no singularities in its interior. Given a regular point $p$, a \\emph{regular closed geodesic} through $p$ is a closed geodesic not passing through any singular points. Regular closed geodesics appear in families of parallel geodesics of the same length, which fill a cylindrical subset of the surface.\n\n\\subsubsection{Holonomy vectors}\\label{subsubsec:holonomy} Let $\\gamma$ be an (oriented) saddle connection or regular closed geodesic. Define the associated holonomy vector \n\\begin{equation}\\label{eq:sc:def} \\mathbf v_{\\gamma} : = \\int_{\\gamma} \\omega.\\end{equation}\n\n\\noindent Note that if $\\gamma$ is a closed geodesic, $\\mathbf v_{\\gamma}$ only depends on the cylinder it is contained in, since regular closed geodesics appearing in a fixed cylinder all have the same length and direction. View $\\mathbf v_{\\gamma}$ as an element of $\\bb R^2$ by identifying $\\bb C$ with $\\bb R^2$. \n\nLet \\begin{eqnarray}\\label{eq:la:def}\\Lambda^{sc}_{\\omega} &=& \\{ \\mathbf{v}_{\\gamma}: \\gamma \\mbox{ a saddle connection on } \\omega\\}\\\\ \\Lambda^{cyl}_{\\omega} &=& \\{ \\mathbf{v}_{\\gamma}: \\gamma \\mbox{ a cylinder on } \\omega\\} \\nonumber\\end{eqnarray} \n\n\\noindent be the set of holonomy vectors of saddle connections and cylinders respectively. For $\\Lambda_{\\omega} = \\Lambda^{sc}_{\\omega}$ or $\\Lambda^{cyl}_{\\omega}$, we have that $\\Lambda_{\\omega}$ is discrete in $\\bb R^2$ (see, e.g., ~\\cite[Proposition 3.1]{Vorobets96}), but Masur~\\cite{Masur:billiards} showed that associated set of directions\n$$\\Theta^{\\omega} : = \\{\\arg(\\mathbf v): \\mathbf v \\in \\Lambda_{\\omega}\\}$$\n\\noindent is dense in $[0, 2\\pi)$ for any $\\omega \\in \\Omega_g$.\n\n\n\n\\subsection{Decay of gaps}\\label{subsec:gap:decay}\nIn this paper, we give a measure of the quantitative nature of this density by considering fine questions about the \\emph{distribution} of saddle connection directions. Given $R >0$, let \n\\begin{equation}\\label{eq:Theta:def} \\Theta^{\\omega}_R : = \\{\\arg(\\mathbf v): \\mathbf v \\in \\Lambda_{\\omega} \\cap B(0, R)\\}\\end{equation} denote the set of directions of saddle connections (or cylinders) of length at most $R$. We write $\\Theta^{\\omega}_R : = \\{0 \\le \\theta_1 <\\theta_2 < \\ldots < \\theta_n<2\\pi\\}$, where $n = \\tilde{N}(\\omega, R)$ is the cardinality of $\\Theta^{\\omega}_R$, and we view $\\theta_{n+1}$ as $\\theta_1$. Note that if we define $$N(\\omega, R) : = |\\Lambda_{\\omega} \\cap B(0, R)|,$$ we have $$\\tilde{N}(\\omega, R) \\le N(\\omega, R).$$ Let $\\gamma^{\\omega}(R)$ be the size of the smallest gap, that is $\\gamma^{\\omega}(R) = \\min_{\\theta_i \\in \\Theta_R} |\\theta_i - \\theta_{i+1}|$, Masur~\\cite{Masur} showed that the counting function $N(\\omega, R)$ grows quadratically in $R$ for any $\\omega$. Since there are at most finitely many (at most $4g-4$) saddle connections in a given direction, this shows that $\\tilde{N}(\\omega, R)$ also has quadratic growth. Thus, one would expect the $\\gamma^{\\omega}(R)$ to decay quadratically. Our main theorem addresses the asymptotic behavior of the rescaled quantity $R^2 \\gamma^{\\omega}(R)$. Let $\\mathcal H$ be a stratum of $\\Omega_g$, and let $\\mu = \\mu_{\\mathcal H}$.\n\n\\begin{Theorem}\\label{theorem:main:gap} For $\\mu$-almost every $\\omega \\in \\mathcal H$,\n\\begin{equation}\\label{eq:main:gap}\\lim_{R \\rightarrow \\infty} R^2 \\gamma^{\\omega}(R) = 0.\n\\end{equation}\n\\noindent Moreover, for any $\\epsilon >0$, the proportion of gaps less than $\\epsilon\/R^2$ is positive. \n\\begin{equation}\\label{eq:main:proportion}\n\\lim_{R \\rightarrow \\infty} \\frac{ |\\{1 \\le i \\le \\tilde{N}(\\omega, R): (\\theta_{i+1} - \\theta_i) \\le \\epsilon\/R^2\\}|}{\\tilde{N}(\\omega, R)} >0.\n\\end{equation}\n\\end{Theorem}\n\n\\bigskip\n\n\\noindent Theorem~\\ref{theorem:main:gap} cannot be extended to \\emph{all} $\\omega \\in \\mathcal H$, since for any stratum $\\mathcal H$ there are many examples $\\omega \\in \\mathcal H$ for which \n\\begin{equation}\\label{eq:gap:lower}\\liminf_{R \\rightarrow \\infty} R^2 \\gamma^{\\omega}(R) >0.\\end{equation}\n\n\\noindent We say that $\\omega$ has \\emph{no small gaps} (NSG) if (\\ref{eq:gap:lower}) holds. An important motivating example of a surface with NSG is the case of the square torus $(\\bb C\/\\bb Z^2, dz)$. Since there are no singular points, there are no saddle connections, but cylinders are given by integer vectors, and $\\Theta^{\\omega_0}$ then corresponds to rational slopes. It can be shown that $3\/\\pi^2$ is a lower bound for $R^2 \\gamma^{\\omega_0}(R)$ (see, for example~\\cite{BCZ1}). \n\nThe torus is an example of a lattice surface. Recall that $\\omega$ is said to be a \\emph{lattice surface} if the group of derivatives of affine diffeomorphisms of $\\omega$ is a lattice in $SL(2, \\bb R)$ (see \\S\\ref{sec:sl2:strata} for more details). We have:\n\n\\begin{Theorem}\\label{theorem:lattice:nsg} $\\omega$ is a lattice surface if and only if it has no small gaps. \\end{Theorem}\n\\medskip\n\nWe prove Theorem~\\ref{theorem:lattice:nsg} in \\S\\ref{subsec:lattice}, using a result of Smillie-Weiss~\\cite{nosmalltri} which characterizes lattice surfaces using the \\emph{no small triangles} (NST) property defined by Vorobets~\\cite{Vorobets96}. \n\nTheorem~\\ref{theorem:main:gap} will follow from a precise statement about the asymptotic distribution of saddle connection directions. This generalizes work of Vorobets~\\cite{Vorobets}, who showed that the sets $\\Theta^{\\omega}_R$ become uniformly distributed (as $R \\rightarrow \\infty$) in $[0, 2\\pi)$ for almost every $\\omega$ (in particular, for those with exact quadratic asymptotics of saddle connections). Our techniques are inspired by those of Marklof-Strombergsson~\\cite{MS}, who studied the distribution of affine lattice points in Euclidean spaces by reducing them to equidistribution problems in homogeneous spaces. Much of the technical machinery is drawn from~\\cite{EskinMasur}, in which Eskin-Masur give precise quadratic asymptotics of $N(\\omega, \\bb R)$ using equidstribution of translate of orbits under the $SL(2, \\bb R)$-action on $\\Omega_g$.\n\n\\subsubsection{Quadratic differentials}\\label{subsubsec:qd} For notational convenience, we work with the space $\\Omega_g$ instead of the space of quadratic differentials $Q_g$. A quadratic differential determines a flat metric, and so saddle connections and cylinders are well defined. For a saddle connection or cylinder curve $\\gamma$, the holonomy vector $\\mathbf v_{\\gamma}$ is given by integrating a square root of the differential, and are thus defined up to a choice of sign. The set of directions can then be viewed as a subset of $[0, \\pi)$. Our results apply, \\emph{mutatis mutandis}, except when explicitly indicated, to the setting of quadratic differentials.\n\n\\subsection{Hyperbolic angle gaps} Higher-genus translation surfaces can be viewed as an intermediate setting betwen flat tori and hyperbolic surfaces. Recently, Boca-Pasol-Popa-Zaharescu~\\cite{BPPZ} study a spiritually similar problem in the hyperbolic setting. They calculate the limiting gap distribution for the angles (measured from the vertical geodesic) of hyperbolic geodesics connecting $i \\in \\bb H^2$ to points in its $SL(2, \\bb Z)$-orbit. In this setting, they show the limiting distribution does have support at $0$, similar to the case of a generic translation surface.\n\n\n\n\n\n\\subsection{Acknowledgements}\\label{subsec:ack} This paper was inspired by \nthe beautiful paper~\\cite{MS} on the distribution of affine \nlattice points. We thank Alex Eskin, Jens Marklof and William Veech for \nuseful discussions. Howard Masur not only patiently answered many technical questions about this paper but more generally taught us much of what we know about the subject. The initial discussions for the project took place while the authors were attending the Hausdorff Institute of Mathematics `Trimester Program on Geometry and Dynamics of Teichm\\\"uller Spaces' in Bonn. We would like to thank the Hausdorff Institute and the organizers of this program for their hospitality. The second author would like to thank the University of Illinois at Urbana-Champaign for its hospitality. The second author was supported in part by an NSF postdoc.\n\\section{The $SL(2, \\bb R)$-action and strata}\\label{sec:sl2:strata}\n\nIn this section, we describe the $SL(2, \\bb R)$ action and stratification of $\\Omega_g$ (\\S\\ref{subsec:polygons} - \\S\\ref{subsec:sl2:affine}), and the construction of an $SL(2, \\bb R)$-invariant measure (\\S\\ref{subsec:coord:meas}). We also describe (\\S\\ref{subsubsec:billiards:translation}) the connection between rational billiards and translation surfaces. This is standard background material in the subject, and our exposition is brief, and drawing on~\\cite{EMM, EMZ}. Excellent general references are~\\cite{Zorich:survey, MasurTab}.\n\n\\subsection{Translation surfaces and polygons}\\label{subsec:polygons} A more geometric description of a translation surface can be given by a union of polygons $P_1 \\cup \\dots \\cup P_n$ where each $P_i \\subset \\bb C$, and the $P_i$ are glued along parallel sides, such that each side is glued to exactly one other, and the total angle in each vertex is an integer multiple of $2 \\pi$. Since translations are holomorphic, and preserve $dz$, we obtain a complex structure and a holomorphic differential on the identified surface. The zeroes of the differential will be at the identified vertices with total angle greater than $2\\pi$. The sum of the excess angles (that is, the orders of the zeros) will be $2g-2$, where $g$ is the genus of the identified surface.\n\n\n\\subsection{Combinatorics of flat surfaces}\\label{subsec:comb:flat} The space $\\Omega_g$ can be stratified by integer partitions of $2g-2$. If $\\alpha =\n(\\alpha_1, \\dots, \\alpha_k)$ is a partition of $2g-2$, we denote by\n$\\mathcal H(\\alpha) \\subset \\Omega_g$ the moduli space of translation surfaces $(M,\\omega)$\nsuch that the multiplicities of the zeroes of $\\omega$ are given by\n$\\alpha_1, \\dots, \\alpha_n$ (or equivalently such that the orders of\nthe conical singularities are $2 \\pi (\\alpha_1 + 1), \\dots, 2\n\\pi(\\alpha_n + 1)$). For technical reasons, the\nsingularities of $(M,\\omega)$ should be labeled; thus, an element of\n$\\mathcal H(\\alpha)$ is a tuple $(M,\\omega, p_1,\\ldots,p_n)$, where\n$p_1,\\ldots,p_n$ are the singularities of~$M$, and the multiplicity\nof~$p_i$ is~$\\alpha_i$. The moduli space of translation surfaces is\nnaturally stratified by the spaces $\\mathcal H(\\alpha)$; each is called a\n{\\em stratum}. Strata are not always connected, but Kontsevich-Zorich~\\cite{KZ} (and Lanneau~\\cite{Lanneau} in the setting of quadratic differentials) have classified the connected components. Most strata are connected, and there are never more than three connected components.\n\n\n\n\n\n\\subsection{$SL(2, \\bb R)$ and affine diffeomorphisms}\\label{subsec:sl2:affine}\n\nThere is an action of $SL(2,\\bb R)$ on the moduli space of\ntranslation surfaces that preserves the stratification. Since $SL(2,\\bb R)$ acts on $\\bb C$ via linear maps on $\\bb R^2$, given a surface $P_1 \\cup \\dots \\cup P_n$, we can define $g S = g P_1 \\cup \\dots \\cup\ng P_n$, where all identifications between the sides of the polygons for\n$gS$ are the same as for $S$. This action generalizes the action of\n$SL(2,\\bb R)$ on the space of (unit-area) flat tori $SL(2,\\bb R)\/SL(2,\\bb Z)$. Note that $SL(2, \\bb R)$ preserves the area of the surface $\\omega$.\n\n\n\\subsubsection{Lattice surfaces}\\label{subsubsec:lattice}\nFor $\\omega \\in \\mathcal H(\\alpha)$, let $\\Gamma(\\omega) \\subset SL(2,\\bb R)$\ndenote the stabilizer of $\\omega$. The group $\\Gamma(\\omega)$ is called the\n{\\em Veech group} of $S$. If $\\Gamma(\\omega)$ is a lattice in $SL(2,\\bb R)$\nthen $\\omega$ is called a {\\em lattice surface}.\n\nEquivalently, let $\\mbox{Aff}(\\omega)$ denote the set of affine (area-preserving) diffeomorphisms of $\\omega$. The derivative of any $f \\in \\mbox{Aff}(\\omega)$ will be a matrix in $SL(2, \\bb R)$, and the collection $\\{Df: f \\in \\mbox{Aff}(\\omega)\\}$ coincides with $\\Gamma(\\omega)$ (up to a finite index subgroup). Thus $\\Gamma(\\omega)$ is a lattice if and only if $D(\\mbox{Aff}(\\omega))$ is.\n\n\\subsubsection{Billiards and translation surfaces}\\label{subsubsec:billiards:translation} An important motivation for studying translation surfaces is their relationship to rational billiards. Recall that a polygon $P \\subset \\bb C$ is called rational if all angles of $P$\nare rational multiples of $\\pi$. The \\emph{unfolding} procedure in ~\\cite{Zelmjakov:Katok} describes how to associate a translation surface $\\omega_P$ so that the billiard flow on $P$ is described by the geodesic flow on $\\omega_P$. \n\nLet $\\Delta_P \\subset O(2)$ denote the group generated by reflections in the sides of the polygon $P$. Since $P$ is rational, $\\Delta_P$ is finite. $\\omega_P$ consists of $|\\Delta_P|$ copies of $P$, with each copy glued to each of its mirror images along the reflecting side.\n\nFor example, if $P$ is the unit square, then $\\omega_P \\in \\mathcal H(\\emptyset)$ is the torus\n$\\bb C\/2 \\bb Z \\oplus 2 \\bb Z$, and if $P$ is the $(\\pi\/8, 3\\pi\/8)$ right triangle, $\\omega_P \\in \\mathcal H(2)$ is a regular octagon with opposite sides identified.\n\n\\subsection{Coordinates and measure on strata}\\label{subsec:coord:meas}\n\nLet $\\alpha = (\\alpha_1, \\ldots, \\alpha_k)$ be an integer partition of $2g-2$. We describe how to put a topology and measure on $\\mathcal H(\\alpha)$. Our exposition is drawn from~\\cite{EMZ}. For a flat surface\n$\\omega_0 \\in\\mathcal H(\\alpha)$ with zero set $\\Sigma = \\{p_1,\\dots, p_k\\}$, choose a\nbasis for the relative homology\n$H_1(\\Sigma_g, \\Sigma;\\bb Z)$. We can pick a basis consisting of saddle connections, since we can choose saddle connections that cut $\\omega_0$ into a\nunion of polygons. For any $\\omega$ near $\\omega_0$ holonomy vectors $\\{\\mathbf v_{\\gamma_i}\\}$ yield local coordinates. That is, we view $\\omega$ as an element of the relative cohomology\n$H^1(\\Sigma_g,\\Sigma;\\bb C) \\cong \\bb R^{4g+2k-2}$, and a domain in this vector space gives us a local coordinate chart. We write $n = 4g+2k-2$. We normalize Lebesgue measure on $\\bb R^n$ so that the integer lattice $\\bb Z^n \\cong H^1(\\Sigma_g,\\Sigma; \\bb Z[i])$ has covolume $1$. Our measure $\\mu(S)$ on $\\mathcal H(\\alpha)$ is given by pulling back this measure via our coordinate maps. This is well-defined, the choice of volume element on $H^1(\\Sigma_g,\\Sigma;\\bb C)$ is independent of choice of basis.\n\nWe will work with \\emph{unit-area} surfaces. Let $\\mathcal H_1(\\alpha)\\subset\\mathcal H(\\alpha)$ be the subset of unit area translation surfaces. Let $a_i,b_i$, $i=1,\\ldots,g$ be a symplectic basis for homology $H_1(\\Sigma_g, \\bb Z)$. The area of the translation surface in the flat metric given by $\\omega$ is given by\n\n$$\\int_{\\Sigma_g} |\\omega|^2 dx dy=\\frac{i}{2}\\int_{\\Sigma_g}\n\\omega\\wedge\\bar\\omega=\n\\frac{i}{2}\\sum_i\\left(\\int_{A_i}\\omega\\int_{B_i}\\bar\\omega-\n\\int_{A_i}\\bar\\omega\\int_{B_i} \\omega\\right).\n$$\n\\noindent This can be viewed as an (indefinite) quadratic form in our local coordinates, and so the level set $\\mathcal H_1(\\alpha)$ can be thought of as a `hyperboloid'. The measure on $\\mathcal H(\\alpha)$ induces a measure on the hypersurface $\\mathcal H_1(\\alpha)$. We can represent any\n$\\omega \\in\\mathcal H(\\alpha)$ as $\\omega = r \\omega'$, where $r\\in\\bb R_+$, and $\\omega' \\in \\mathcal H_1(\\alpha)$.\nHolonomy vectors of saddle connections and cylinders on $\\omega'$ are\nmultiplied by $r$ to give vectors associated to corresponding\nsaddle connections on $\\omega$, and $\\mbox{area}(\\omega) =\nr^2\\cdot\\mbox{area}(\\omega')=r^2$. The measure $\\mu_1$ on\n$\\mathcal H_1(\\alpha)$ is given by disintegration of the volume element $\\mu$\non $\\mathcal H(\\alpha)$:\n %\n$$\nd\\mu(\\omega) = r^{n-1} \\, dr\\, d\\mu_1(\\omega').\n$$\n\n\\noindent In the sequel, by abuse of notation, we will fix a connected component $\\mathcal H$ of $\\mathcal H_1(\\alpha)$ and denote the Lebesgue measure on it by $\\mu$. We note that in any stratum, the set of surfaces arising from billiards as in \\S\\ref{subsubsec:billiards:translation} has measure zero. Thus, statements about almost every translation surface do not yield results about billiard flows, in particluar, Theorem~\\ref{theorem:main:gap} does not apply to billiard flows. In \\S\\ref{sec:billiards} we discuss some special classes of billiards for which we can prove a version of Theorem~\\ref{theorem:main:gap}.\n\n\\subsubsection{$SL(2,\\bb R)$-invariance and ergodicity}\\label{subsubsec:ergodic} Since the $SL(2,\\bb R)$-action on $\\mathcal H(\\alpha)$ preserves the area, it acts on the level set $\\mathcal H_1(\\alpha)$. It also preserves connected components, so we can consider it acting on $\\mathcal H$. The measure $\\mu$ constructed above is invariant under $SL(2,\\bb R)$. The following theorem is due (independently) to Veech~\\cite{Veech:gauss} and Masur~\\cite{Masur:IET}.\n\n\\begin{theorem*}[Veech~\\cite{Veech:gauss}, Masur~\\cite{Masur:IET}] $\\mu$ is a finite, ergodic, $SL(2,\\bb R)$-invariant measure on $\\mathcal H$. \\end{theorem*}\n\n\\subsubsection{Short saddle connections}\\label{subsubsec:short:sc} We record here a crucial measure estimate on the set of surfaces with short saddle connections. It is originally due to Masur-Smillie~\\cite{MasurSmillie} we recall it as it is quoted in~\\cite[Lemma 7.1]{EMZ}\n\n\\begin{lemma}\n[H.~Masur, J.~Smillie]\n\\label{lemma:short:saddle:connections}\n %\nThere is a constant $M$ such that for all $\\epsilon,\\kappa>0$ the\nsubset of $\\mathcal H$ consisting of those flat surfaces,\nwhich have a saddle connection of length at most $\\epsilon$, has\nvolume at most $M\\epsilon^2$. The volume of the set of flat\nsurfaces with a saddle connection of length at most $\\epsilon$\nand a nonhomologous saddle connection with length at most\n$\\kappa$ is at most $M\\epsilon^2\\kappa^2$.\n %\n\\end{lemma}\n\n\\noindent\\textbf{Remark:} Note that by the construction of the measure, the volumes of these sets will be \\emph{at least} $m \\epsilon^2$ and $m \\epsilon^2 \\kappa^2$ for some possibly smaller $m$, since we can construct local coordinates using a basis given by our short saddle connections.\n\n\\section{Saddle connections}\\label{sec:results}\n\n\\noindent This section contains statements of our main results. In \\S \\ref{subsec:windows}, we give our main distribution result Theorem~\\ref{theorem:wedge:ae} for the directions of saddle connections and cylinders. In \\S\\ref{subsec:proof:main:gap} we show how to use this result to derive Theorem~\\ref{theorem:main:gap}. Theorem~\\ref{theorem:wedge:ae} relies on results on limit measures for certain subsets of $SL(2, \\bb R)$-orbits, which we describe in \\S\\ref{subsec:limit:circles}, and show how to use these limit theorems to obtain results for billiards. In \\S\\ref{subsec:lattice}, we describe how lattice surfaces yield exceptional behavior in our context. In \\S\\ref{subsec:limit:dist} we state results on measure bounds and gap distribution.\n\n\n\\subsection{Counting points in thinning segments}\\label{subsec:windows} \n\nWe recall notation: $\\mathcal H$ is a connected component of a stratum $\\mathcal H_1(\\alpha)$ of unit area differentials in $\\Omega_g$, and $\\mu$ is Lebesgue measure on $\\mathcal H$, normalized to be a probability measure. Given $\\omega \\in \\mathcal H$, $\\Lambda_{\\omega}$ denotes set of holonomy vectors of either saddle connections or periodic cylinders in the flat metric determined by $\\omega$. \n\n\\subsubsection{The Siegel-Veech transform}\\label{subsubsec:siegel:veech} We recall the defintion of the \\emph{Siegel-Veech transform} from~\\cite[\\S2.1]{EskinMasur}. Given a compactly supported function $f: \\bb R^2 \\rightarrow \\bb R$, define $\\hat{f}: \\mathcal H \\rightarrow \\bb R$ by\n\\begin{equation}\\label{eq:sv:transform}\\hat{f}(\\omega) = \\sum_{\\mathbf v \\in \\Lambda_{\\omega}} f(\\mathbf v).\n\\end{equation}\n\nVeech~\\cite{Veech} formulated the following seminal result, now known as the \\emph{Siegel-Veech formula}:\n\n\\begin{Theorem}\\label{theorem:sv:formula} Let $\\eta$ be an ergodic $SL(2, \\bb R)$ invariant probability measure on $\\mathcal H$. There is a $b = b(\\eta)$ so that for all $f \\in C^{\\infty}_0(\\bb R^2)$, $$\\int_{\\mathcal H} \\hat{f} d\\eta = b \\int_{\\bb R^2} f dm$$\n\\end{Theorem}\n\n\n\\subsubsection{Thinning annular regions}\\label{subsec:annular} Following \\cite[\\S 2.3]{MS}, we consider a family of thinning annular regions in $\\bb R^2$ (see Figure~\\ref{polar} below): given $\\theta \\in [0, 2\\pi)$, $\\sigma, R >0$, and $0 \\le c < 1$ define the annular region\n\\begin{equation}\\label{eq:c:theta:sigma} \nA^{\\theta}_R(c, \\sigma) : = \\{\\mathbf v \\in \\bb R^2: cR \\le ||\\mathbf v|| \\le R, \\arg(\\mathbf v) \\in (\\theta - \\sigma R^{-2}, \\theta + \\sigma R^{-2})\\}.\n\\end{equation}\n\\makefig{The region $A = A^{\\theta}_R(c, \\sigma)$}{polar}{\\input{wedge.pstex_t}} \n\n\n\n\\noindent As $R \\rightarrow \\infty$, this gives a narrowing wedge of directions around the angle $\\theta$. For $\\omega \\in \\mathcal H$ define the counting function \n\n\\begin{equation}\\label{eq:n:theta:sigma} N^{\\theta}_R(\\omega, \\sigma, c) : = |\\Lambda_{\\omega} \\cap A^{\\theta}_R(c, \\sigma) |.\\end{equation}\n\n\\noindent We think of $N^{\\theta}_R(\\omega, \\sigma, c)$ as the number of saddle connections in a small neighborhood of the direction $\\theta$. Note that $N^{\\theta}_R(\\omega, \\sigma, c)$ can be viewed as the Siegel-Veech transform of the indicator function of $A^{\\theta}_R(c, \\sigma)$. Given that $|\\Lambda_{\\omega} \\cap B(0, R)|$ has quadratic asymptotics, one would expect $N^{\\theta}_R(\\omega, \\sigma)$ to be proportional to $\\sigma$. We frame the following question. Fixing an integer $k$, if we choose $\\theta$ uniformly in $[0, 2\\pi)$, what is the probability that $N^{\\theta}_R(\\omega, \\sigma, c) = k$?\n\n\\begin{Theorem}\\label{theorem:wedge:ae} Fix $\\sigma >0$ and $c \\in [0, 1)$, $k \\in \\bb Z_{\\geq 0}$. Let $\\mathcal H$ be a (connected component of) stratum $\\mathcal H_1(\\alpha)$ and let $\\mu = \\mu_{\\mathcal H}$ denote the natural $SL(2,\\bb R)$-invariant probability measure on $\\mathcal H$. For $\\mu$-a.e. $\\omega_0 \\in \\mathcal H$, \n\\begin{equation}\\label{eq:wedge:ae}\n\\lim_{R \\rightarrow \\infty} \\lambda(\\theta: N^{\\theta}_R(\\omega_0, \\sigma, c) = k) = \\mu(\\omega: \\Lambda_{\\omega} \\cap T(c, \\sigma) = k)\\end{equation}\n\n\\noindent where $\\lambda$ denotes the Lebesgue probability measure on $[0, 2\\pi)$ and $T(c, \\sigma)$ is the trapezoid with vertices $(c, \\pm c\\sigma), (1, \\pm \\sigma)$.\n\n\\end{Theorem}\n\n\\medskip\n\n\\noindent In \\S\\ref{sec:axiom}, we will see that this theorem will hold with $\\Lambda_{\\omega}$ replaced by other sets of holonomy vectors of special trajectories on $\\omega$.\n\n\\subsection{Proof of Theorem~\\ref{theorem:main:gap}}\\label{subsec:proof:main:gap} We show how Theorem~\\ref{theorem:main:gap} follows from Theorem~\\ref{theorem:wedge:ae}. Given $k \\in \\bb Z_{\\geq 0}, \\sigma >0$, let\n\\begin{equation}\\label{eq:pksigma:def} p_k(\\sigma) := \\mu(\\omega: \\Lambda_{\\omega} \\cap T(\\sigma) = k)\\end{equation}\n\\noindent where $T(\\sigma) = T(0, \\sigma)$. We require the following lemma.\n\n\\begin{lemma}\\label{lemma:p2sigma} For any $\\sigma>0$\n\\begin{equation}\\label{eq:p2sigma} p_{2}(\\sigma)>0. \\end{equation}\n\n\\end{lemma}\n\n\\noindent\\textbf{Proof:} As in the remark following Lemma~\\ref{lemma:short:saddle:connections}, we note that we can construct in $\\mathcal H$ a set of measure at least $m \\sigma^4$ ($m$ depending on $\\mathcal H$) with two saddle connections with holonomy vectors in $T(\\sigma)$.\n\\qed\n\\medskip\n\n\\noindent Let $n \\in \\bb N$, let $\\mathcal H_{n} \\subset \\mathcal H$ be the full measure set of $\\omega \\in \\mathcal H$ so that (\\ref{eq:wedge:ae}) holds for $\\sigma = 1\/n$, $c =0$. Then $\\mathcal H_{\\infty} = \\bigcap_{n=1}^{\\infty} \\mathcal H_n$ is also a full measure set. We claim that for any $\\omega_0 \\in \\mathcal H_{\\infty}$, (\\ref{eq:main:gap}) holds, that is, \n$$\\lim_{R \\rightarrow \\infty} R^2 \\gamma^{\\omega_0}(R) = 0.$$\n\\noindent Let $n \\in \\bb N$. Since $\\omega_0 \\in \\mathcal H_{\\infty}$, (\\ref{eq:wedge:ae}), there is an $R_n$ such that \nfor all $R> R_n$, \n\\begin{equation}\\label{eq:lambda:positive}\n\\lambda(\\theta: N^{\\theta}_R(\\omega_0, 1\/n, 0) \\geq 2) \\geq p_2(\\sigma)\/2 >0. \\end{equation}\n\\noindent The last inequality follows from Lemma~\\ref{lemma:p2sigma}. By construction of $N^{\\theta}_R(\\omega_0, 1\/n, 0)$, (\\ref{eq:lambda:positive}) implies that for $R > R_n$, \n$$\\gamma^{\\omega_0}(R) \\le \\frac{1}{nR^2}.$$\n\\noindent Since $n$ was arbitrary (\\ref{eq:main:gap}) follows. To see (\\ref{eq:main:proportion}), note that $\\lambda(\\theta: N^{\\theta}_R(\\omega_0, \\epsilon, 0) \\geq 2)$ gives a lower bound for the limiting proportion of gaps of size less than $\\epsilon\/R^2$. Theorem \\ref{theorem:main:gap} now follows from Theorem~\\ref{theorem:wedge:ae} and Lemma~\\ref{lemma:p2sigma}. \\qed\\medskip\n\n\\subsection{Limit measures}\\label{subsec:limit:circles} The main weakness of Theorem~\\ref{theorem:wedge:ae} is that it does not give us information about any particular surface $\\omega_0 \\in \\mathcal H$. To obtain such information, we must have further knowledge about the limiting behavior (as $t \\rightarrow \\infty$) of the orbits $\\{g_t r_{\\theta} \\omega_0: 0 \\le \\theta < 2\\pi\\}$, where \n\n \\begin{equation}\n \\label{eq:matrices}\ng_t = \\left(\\begin{array}{cc} e^{-t\/2} & 0 \\\\ 0 & e^{t\/2}\n\\end{array}\\right),\nr_{\\theta} = \\left(\\begin{array}{cc} \\cos\\theta& -\\sin\\theta \\\\ \\sin\\theta & \\cos\\theta\n\\end{array}\\right).\n\\end{equation}\n\n\\noindent Let $\\nu_{t, \\omega_0}$ denote the Lebesgue probability measure supported on $\\{g_t r_{\\theta} \\omega_0: 0 \\le \\omega_0 < 2\\pi\\}$. Suppose $\\lim_{t \\rightarrow \\infty} \\nu_{t, \\omega_0} = \\mu_0$, and $\\mu_0$ is $SL(2, \\bb R)$-invariant. In this case we say $\\mu_0$ is the \\emph{circle limit measure} associated to $\\omega_0$. By~\\cite[Theorem 5.2]{EskinMasur}, $\\mu_0$ is a probability measure. \n\n\\begin{Theorem}\\label{theorem:wedge:limit} Suppose $\\omega_0 \\in \\mathcal H$ has circle limit measure $\\mu_0$. Then \\begin{equation}\\label{eq:wedge:limit}\n\\lim_{R \\rightarrow \\infty} \\lambda(\\theta: N^{\\theta}_R(\\omega_0, \\sigma, c) = k) = \\mu_0(\\omega: \\Lambda_{\\omega} \\cap T(c, \\sigma) = k).\\end{equation}\n\\end{Theorem}\n\n\\medskip \n\n\\noindent Letting $\\gamma^{0}(R) = \\gamma^{\\omega_0}(R)$, the proof of Theorem~\\ref{theorem:main:gap} yields the following corollary to Theorem~\\ref{theorem:wedge:limit}. Let $p^0_2(\\sigma) : = \\mu_0(\\omega: \\Lambda_{\\omega} \\cap T(\\sigma) = 2)$.\n\n\\begin{Cor}\\label{cor:gap:limit} Fix notation as in Theorem~\\ref{theorem:wedge:limit}. Suppose for all $\\sigma >0$, $p^0_2(\\sigma)>0$. Then \n\\begin{equation}\\label{eq:gap:limit}\\lim_{R \\rightarrow \\infty} R^2 \\gamma^{0}(R) = 0.\n\\end{equation}\n\\noindent Moreover, for any $\\epsilon >0$, the proportion of gaps less than $\\epsilon\/R^2$ is positive. That is, writing $\\Theta^{\\omega_0}_R : = \\{0 \\le \\theta_1 \\le \\theta_2 \\le \\ldots \\le \\theta_n\\}$, we have\n\\begin{equation}\\label{eq:limit:proportion}\n\\lim_{R \\rightarrow \\infty} \\frac{ |\\{1 \\le i \\le \\tilde{N}(\\omega_0, R): (\\theta_{i+1} - \\theta_i) \\le \\epsilon\/R^2\\}|}{\\tilde{N}(\\omega_0, R)} >0.\n\\end{equation}\n\\end{Cor}\n\n\\medskip\n\n\\subsubsection{Billiards with barriers}\\label{subsubsec:billiards:barriers} We describe how Theorem~\\ref{theorem:wedge:limit} and Corollary~\\ref{cor:gap:limit} can be used to give information about specific families of billiards. Following~\\cite{EMS}, we consider the following family of billiards. Given $\\alpha \\in \\bb R$, consider the polygon $P_{\\alpha}$ whose boundary is the boundary of the square $[0, 1] \\times [0, 1]$ together with a barrier given by the vertical segment $\\{1\/2\\} \\times [0, \\alpha]$. \n\n\\medskip\n\n\\noindent\\textbf{Remark:} In fact, slits based at any rational point $p\/q$ are considered in~\\cite{EMS}. We restrict to $1\/2$ for notational convenience and ease of exposition.\n\n\\medskip\n\nThe associated surface, which we denote by $\\omega_{\\alpha}$ is (after rescaling) an element of $\\mathcal H(1,1)$. A crucial observation is that $\\omega_{\\alpha}$ is a double cover of the torus. That is, there is a covering map (branched at the zeros) $\\pi: \\omega_{\\alpha} \\rightarrow \\bb C\/\\Lambda$ where $\\Lambda$ is a lattice in $\\bb C$ and that $\\omega_{\\alpha}$ is obtained by pulling back the form $dz$. Using this construction, we observe that the set $\\{\\omega_{\\alpha}: \\alpha \\in \\bb R\\}$ is contained in an $SL(2,\\bb R)$-invariant subvariety $\\mathcal M \\subset \\mathcal H$ which can be identified with the moduli space of tori with marked points, $(SL(2,\\bb R)\\ltimes\\bb R^2)\/(SL(2,\\bb Z)\\ltimes\\bb Z^2)$. \n\nLet $\\mu_{\\mathcal M}$ denote the natural $SL(2,\\bb R)$-invariant probability measure supported on $\\mathcal M$. Applying a theorem of Shah~\\cite{Shah:SL2} (which uses Ratner's measure classification), it is shown in~\\cite{EMS} that for any irrational $\\alpha$, $\\mu_{\\mathcal M}$ is a circle limit measure for $\\omega_{\\alpha}$. We will see in \\S\\ref{subsec:translate} that for any $\\sigma >0$,\n$$p^{\\mathcal M}_2(\\sigma) := \\mu_{\\mathcal M}(\\omega: \\Lambda_{\\omega} \\cap T(\\sigma) = 2)>0,$$\n\n\\noindent and thus, Corollary~\\ref{cor:gap:limit} applies in this situation.\n\n\n\n\\subsection{Lattice surfaces}\\label{subsec:lattice} In this section we prove Theorem~\\ref{theorem:lattice:nsg}. Fix $\\gamma^{\\omega}_R$ to denote the smallest gap for saddle connections of length at most $R$. We split the proof into two lemmas. The first shows that any lattice surface has NSG.\n\n\\begin{lemma}\\label{lemma:lattice:gap} For any lattice surface $\\omega$ there exists a constant $\\epsilon>0$ such that $\\gamma^{\\omega}_R\\geq \\frac{\\epsilon}{R^2}$ for all $R>1.$\n\\end{lemma}\n\\begin{proof}We want to show that if $\\omega \\in \\mathcal H$ is a lattice surface, then $\\omega$ has no small gaps. By \\cite[Proposition 6.1]{Vorobets96} if $\\omega$ is a lattice surface there exists a constant $s$ such that any two saddle connections in the same direction have the ratio of their lengths at most $s$.\n Given a periodic direction $\\theta$ there must be a cylinder of area at least $\\frac 1 {4g-4}$ in that direction. Let $R$ be its length. By Lemma \\ref{cyl flow} points in this cylinder are not in another cylinder of length at most $R$ in a direction within $\\frac{1}{(8g-8)R^2}$. Any other saddle connection in this direction must have length $\\frac{R} s$ which implies that a direction $\\psi \\in B(\\theta,\\frac{1}{(8g-8)R^2})$ can have no periodic cylinders of length less than $\\frac R s$. \\end{proof}\n \n \\medskip\n \n\\noindent For the converse, we recall that $\\omega$ has \\emph{no small triangles} if there is a $\\delta >0$ so that all triangles on $\\omega$ with vertices at singularities, and no singularities in the interior have area at least $\\delta$. Smillie-Weiss~\\cite{nosmalltri} showed:\n\n\\begin{theorem*} $\\omega$ is a lattice surface if and only if it has no small triangles. \\end{theorem*}\n\n\\medskip\n\n\\noindent Combining this theorem with the following lemma completes the proof of Theorem~\\ref{theorem:lattice:nsg}. \n\n\\begin{lemma}\\label{lemma:nst:nsg} If $\\omega$ has no small gaps then it has no small triangles.\\end{lemma}\n\\begin{proof} Let $\\epsilon >0$ be such that $R^2 \\gamma^{\\omega}_R > \\epsilon$. Let $T$ be a triangle on $\\omega$ with vertices at singularities and with no singularities in the interior. Without loss of generality we can assume that the sides of $T$ are saddle connections, since if not it can be decomposed into triangles which are. Let $R$ be the length of the longest side. Dropping a perpendicular from the opposite vertex, we decompose the side into two segments, at least one of which has length at least $R\/2$. Consider the right triangle formed by the perpendicular and this segment. The angle opposite the perpendicular is an angle between saddle connections of length at most $R$, so it is at least $\\frac{\\epsilon}{R^2}$. See Figure~\\ref{triangle} below.\n\n \\makefig{$\\theta \\geq \\frac{\\epsilon}{R^2}$}{triangle}{\\input{triangle.pstex_t}} \n\n\n\\noindent The length $L$ of the perpendicular is at least $\\frac{R}{2}\\tan(\\frac{\\epsilon}{R^2})$, so $L > \\frac{R}{2}\\frac{\\delta}{R^2} = \\frac{\\delta}{2R}$ for some $\\delta > 0$. Thus the area of the triangle is bounded below by $\\frac{\\delta}{4}$, and so is the area of $T$. Since $T$ was arbitrary, we have the $\\omega$ has no small triangles.\n\\end{proof}\n\n \n \n\n\n\nIf we replace saddle connections with cylinders, then, as pointed out to us by Barak Weiss, we can construct a non-lattice surface with no small cylinder gaps as follows: take a branched cover of a torus by varying relative periods (see \\S\\ref{sec:billiards}). This surface has absolute holonomy in\n$\\bb Z^2$ and therefore has no small gaps for vectors which are holonomies of\ncylinder core curves. \n\n\\subsection{Measure bounds and gaps}\\label{subsec:limit:dist}\nLet $\\hat{G}_2(\\Theta_R^{\\omega})=\\left\\{R^2 (\\theta_{i+1}-\\theta_i)\\right\\}_{i=1}^{|\\Theta_R^{\\omega}|}$.\n Let $\\nu_2^{\\omega}(R)$ be the probability measure obtained by normalizing the measure given by delta mass at\n each element of $\\hat{G}_2(\\Theta_R^{\\omega})$.\n\\begin{Prop}\\label{lim distr} For almost every surface $\\omega$ the measure $\\nu_2^{\\omega}(R)$ converges (as $R \\rightarrow \\infty$) in the weak-* topology.\n\\end{Prop}\n\\begin{proof} The existence of $p_0(\\sigma)$ and the quadratic growth of saddle connections\nimplies this by \\cite[Theorem 2.1]{MS}. In particular, after \nunwinding the definitions there we have \n$\\nu_{\\infty}([a,b])=\\frac {d} {d\\sigma} p_0(\\sigma)|_{a}-\\frac d {d\\sigma}p_0(\\sigma)|_b$. \nNote that this makes sense for almost every $\\sigma$ because $p_0(\\sigma)$ is a decreasing function of $\\sigma$.\n\\end{proof}\n\n\n\n\n\\section{Equidistribution on strata}\\label{sec:axiom}\n\nIn this section, we prove Theorem~\\ref{theorem:wedge:ae} and and Theorem~\\ref{theorem:wedge:limit}. We follow the strategy outlined in~\\cite{EskinMasur} for proving results on the asymptotics of $N(\\omega, R)$ and modify the techniques to our situation.\n\n\\subsection{Cones in $\\bb R^2$}\\label{subsec:cones}\n\nThe crucial geometric observation in the proof of Theorems~\\ref{theorem:wedge:ae} and~\\ref{theorem:wedge:limit} is the following. Let $R>>0$, and $t = 2 \\log R$. Then \n$$r_{\\theta} g_{-t} T(c, \\sigma) \\approx A^{\\theta}_R(c, \\sigma)$$\n\\noindent This can be seen as follows: $g_{-t} T(c, \\sigma)$ is a trapezoid with vertices at $(cR, \\pm c \\sigma \/R), (R, \\pm \\sigma\/ R)$. Rotating it by angle $\\theta$, we have that $r_{\\theta} g_{-t} T(c, \\sigma)$ is a thin trapezoidal wedge around the set line $\\{\\mathbf v \\in \\bb R^2: \\arg(\\mathbf v) = \\theta\\}$, with vectors of length roughly between $cR$ and $R$ (there is an error of up to $ \\frac 1 R$ because $r_{\\theta}g_{-t}T(c,\\sigma)$ is a trapezoid and not a wedge of an annulus). Finally, the angular width of this wedge is roughly $c\/R^2$, since for small angles $\\phi$ the slope $\\tan(\\phi) \\approx \\phi$. Thus, for any $\\omega \\in \\mathcal H$, $R >>0$,\n\\begin{equation}\\label{eq:lambda:wedge}|\\Lambda_{\\omega} \\cap A^{\\theta}_R(c, \\sigma)| \\approx |g_{t} r_{-\\theta} \\Lambda_{\\omega} \\cap T(c, \\sigma)|.\\end{equation}\n\n\\noindent For $k \\in \\bb N$, let $f_k: \\mathcal H \\rightarrow [0, 1]$ be the indicator function of the set \n\\begin{equation}\\label{eq:level:set}\\mathcal H_{c, \\sigma, k} : = \\{\\omega \\in \\mathcal H: |\\Lambda_{\\omega} \\cap T(c, \\sigma)| = k\\}.\\end{equation}\n\n\\noindent Using (\\ref{eq:lambda:wedge}), we can write\n\\begin{equation}\\label{eq:circle:relation}\n \\lambda(\\theta: N^{\\theta}_R(\\omega, \\sigma, c) = k) \\approx \\int_0^{2\\pi} f_k(g_t r_{-\\theta} \\omega) d\\lambda(\\theta) = \\int_{\\mathcal H} f_k d\\nu_{t,\\omega}.\n\\end{equation}\n\\noindent Here, $\\approx$ denotes that the difference goes to $0$ as $R \\rightarrow \\infty$. We will rigorously justify (\\ref{eq:circle:relation}) below.\n\\subsection{Equidistribution}\\label{subsec:equi} Equation (\\ref{eq:circle:relation}) reduces proving Theorem~\\ref{theorem:wedge:ae} and Theorem~\\ref{theorem:wedge:limit} to understanding \n$$\\lim_{t \\rightarrow \\infty} \\int_{\\mathcal H} f_k d\\nu_{t,\\omega}.$$We first prove Theorem~\\ref{theorem:wedge:limit} assuming (\\ref{eq:circle:relation}):\n\n\\noindent\\textbf{Proof of Theorem~\\ref{theorem:wedge:limit}:} While there is no general pointwise ergodic theorem known for the measures $\\nu_{t,\\omega}$, our functions $f_k$ are indicator functions of level sets of the Siegel-Veech transform of the indicator function $h$ of $T(c, \\sigma)$ (see \\S\\ref{subsubsec:SV} below). Following~\\cite[\\S4]{EskinMasur}, there is an approximation argument that allows us to conclude that if $\\nu_{t,\\omega_0} \\rightarrow \\mu_0$, we have, as desired\n$$\\lim_{t \\rightarrow \\infty} \\int_{\\mathcal H} f_k d\\nu_{t, \\omega_0} = \\int_{\\mathcal H} f_k d\\mu_0 = \\mu_0(\\omega \\in \\mathcal H: \\Lambda_{\\omega} \\cap T(c, \\sigma) = k).$$\n\\noindent\\qed\\medskip\n\n\n\\noindent\\textbf{Proof of Theorem~\\ref{theorem:wedge:ae}:} Combine Theorem~\\ref{theorem:wedge:limit} with~\\cite[Proposition 3.3]{EskinMasur}.\n\\qed\\medskip\n\n\\subsubsection{Proof of (\\ref{eq:circle:relation})}\\label{sec:circle:relation}\n\nLet $F_t$ denote the indicator function of $g_{t} r_{-\\theta} A^{\\theta}_R(c, \\sigma)$. By construction, this does \\emph{not} depend on $\\theta$, as it is the $g_{t}$ image of a horizontal cone. As above, let $h$ denote the indicator function of $T(c, \\sigma)$. We abbreviate $\\nu_{t, \\omega_0}$ by $\\nu_t$. Denote the symmetric difference of $g_t r_{-\\theta} A^{\\theta}_R(c, \\sigma)$ and $T(c, \\sigma)$ by $E_t$, and note that $E_s \\subset E_t$ for $s > t$, and the volume of $E_t$ tends to zero uniformly in $\\theta$ as $t \\rightarrow \\infty$. Also note that by construction, $$\\widehat{F_t}(g_{t} r_{-\\theta} \\omega) = N^{\\theta}_R(\\omega, \\sigma, c),$$ and $f_k$ is the indicator function of the level set $\\{\\omega: \\widehat{h}(\\omega) = k\\}$. Thus we would like to show that, given any basepoint $\\omega_0$, $$\\lim_{t \\rightarrow \\infty} \\nu_t (\\omega: \\widehat{F_t} (\\omega) = k) = \\lim_{t \\rightarrow \\infty} \\nu_t(\\omega: \\widehat{h}(\\omega) = k).$$ Recall that for any set $L \\subset \\mathcal H$, $$\\nu_t(\\omega: \\omega \\in L) = \\lambda(\\theta: g_{-t} r_{-\\theta} \\omega_0 \\in L).$$\n\n\\noindent Let $I_t = \\{\\omega: \\widehat{F_t}(\\omega) \\neq \\widehat{g}(\\omega)\\}$. Letting $b_0$ denoting the Siegel-Veech constant (see Theorem~\\ref{theorem:sv:formula}) of $\\mu_0$, we have $$\\mu_0(I_t) < b_0 m(E_t),$$where $m$ denotes Lebesgue measure on $\\bb R^2$. Also note that $I_s \\subset I_t$ for $s > t$. Fix $\\epsilon >0$. Let $t_0 >0$ be such that $m(E_t) < \\epsilon\/C_0$ for all $t > t_0, \\theta \\in [0, 2\\pi)$, so $$\\mu_0(I_t) < \\epsilon.$$\n\n\\noindent Since $\\nu_t \\rightarrow \\mu_0$, we can pick $t_1 >t_0$ so that for all $t > t_1$, $$\\nu_t (I_t) \\le \\nu_t(I_{t_0}) \\le 2 \\epsilon.$$\n\n\\noindent Write $$\\nu_t(\\omega: \\widehat{F_t}(\\omega) = k ) = \\nu_t(\\omega \\notin I_t: \\widehat{F_t} = k) + \\nu_t(\\omega \\in I_t: \\widehat{F_t}(\\omega) = k)$$\n\n\\noindent The second term can be bounded above by $\\nu_t(I_t)$ and thus by $2 \\epsilon$. The first term can be rewritten as \n$$\\nu_t(\\omega: \\widehat{g} (\\omega) = k) - \\nu_t(\\omega \\in I_t: \\widehat{g}(\\omega) = k)$$\n\n\\noindent Again using our bound on $\\nu_t(I_t)$, the difference $$|\\nu_t(\\omega: \\widehat{F_t}(\\omega) = k ) -\\nu_t(\\omega: \\widehat{g} (\\omega) = k)|$$ can be bounded by $4 \\epsilon$. Passing to the limit, we obtain that the limits can differ by no more than $4 \\epsilon$. Since $\\epsilon$ was arbitrary, we have that the limits must be equal. \\qed\\medskip\n\n\n\\subsubsection{The Siegel-Veech formula}\\label{subsubsec:SV} Using the Siegel-Veech transform, we can obtain results on the \\emph{expected} number of points in a thinning wedge. We fix some notation. Recall that for a bounded compactly supported function $f: \\bb R^2 \\backslash \\{0\\} \\rightarrow \\bb R$, we define $\\widehat{f}(\\omega) = \\sum_{\\mathbf v \\in \\Lambda_{\\omega}} f(\\mathbf v)$. Fixing $R, c, \\sigma$, let $h_{\\theta} = \\chi_{A^{\\theta}_R(c, \\sigma)}$, that is, it is the indicator function of the thinning wedge. The expected number of lattice points in a random thinning wedge can be written as\n\n$$\\sum_{k=1}^{\\infty} k \\lambda(\\theta: N^{\\theta}_R(\\omega, \\sigma, c) = k) = \\int \\widehat{h_{\\theta}}d\\lambda(\\theta).$$\n\n\\noindent Writing $h$ for the indicator function of $T(c, \\sigma)$, and using (\\ref{eq:lambda:wedge}) we can write this (for $t>>0$) as \n$$\\int_{\\mathcal H} \\widehat{h} d\\nu_{t,\\omega}.$$\n\n\\noindent If $\\omega_0$ has a circle limit measure $\\mu_{0}$ satisfying a certain technical condition~\\cite[Theorem 8.2(D)]{EMM}, then \n$$\\lim_{t \\rightarrow \\infty} \\int_{\\mathcal H} \\widehat{h} d\\nu_{t,\\omega} = \\int_{\\mathcal H} \\widehat{h} d\\mu_0.$$\n\n\\noindent As above, let $b_0$ be the Siegel-Veech constant for $\\mu_0$. Then we have\n\\begin{equation}\\label{eq:siegel:veech}\\int_{\\mathcal H} \\widehat{h} d\\mu_0 = b_0 \\int_{R^2} h.\\end{equation}\n\n\\noindent That is, the expected number of points in a thinning wedge is proportional to the volume of the trapezoid $T(c, \\sigma)$. We will see in \\S\\ref{subsec:second:moment} that the \\emph{second moment} of the limiting distribution is \\emph{not} well-defined.\n\n\\subsubsection{Fiber bundles and special trajectories}\\label{subsubsec:fiber} In this paper, we focus on the sets of holonomy vectors of oriented saddle connections or cylinders. Our results will also apply to the sets of holonomy vectors connecting a fixed point on the surface to singular points or the set of vectors connecting two marked points. These can be obtained by considering the corresponding equidistribution results for the spaces $Y_i$ of translations surfaces with $i$ marked points, $i=1, 2$. These spaces can be viewed as fiber bundles over $\\mathcal H$ with fiber over $\\omega \\in \\mathcal H$ given by $(M, \\omega)^i$. We refer the interested reader to~\\cite[\\S2, \\S9]{EskinMasur} for details.\n\n\\section{Measure bounds and gap distribution}\\label{sec:measure:bounds}\nThis section provides a variety of results on the gaps between saddle connection and cylinder directions. It includes results on the likelihood of finding many saddle connections in a small region. In particular, Lemma \\ref{k asym} says that having $k$ saddle connections in a small interval is proportional to having two saddle connections in a small interval. As a corollaries we show Corollary \\ref{cor:second:moment} which says that the limiting distribution of $p_k(\\sigma)$ does not have finite second moments and Corollary \\ref{SV not L2} which says that the Siegel-Veech transform does not send continuous compactly supported functions to $L_2$ (though it is norm preserving for positive functions as a map from $L_1$ to $L_1$). We also show Theorem \\ref{p sig 0 lower} which says that some gaps between cylinder directions are larger than one would expect .\n\\subsection{Notation}\\label{subsec:not}\n\nIn this section, we will need to distinguish between holonomy vectors of saddle connections and cylinders. Let $\\omega \\in \\mathcal H$, let $\\Lambda^{sc}_{\\omega}$ and $\\Lambda^{cyl}_{\\omega}$ be as in (\\ref{eq:la:def}), and let $\\mu$ be Lebesgue measure on $\\mathcal H$. We define\n\n\\begin{eqnarray}\\label{eq:tildepksigma:def} p_k(\\sigma) :&=& \\mu(\\omega: \\Lambda^{sc}_{\\omega} \\cap T(\\sigma) = k) \\\\ \\nonumber \\tilde{p}_k(\\sigma) :&=& \\mu(\\omega: \\Lambda^{sc}_{\\omega} \\cap T(\\sigma) \\geq k)\\\\ \\nonumber p^{cyl}_k(\\sigma) :&=& \\mu(\\omega: \\Lambda^{cyl}_{\\omega} \\cap T(\\sigma) = k) \\\\ \\nonumber \\tilde{p}^{cyl}_k(\\sigma) :&=& \\mu(\\omega: \\Lambda^{cyl}_{\\omega} \\cap T(\\sigma) \\geq k).\\\\ \\nonumber \\end{eqnarray}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\sigma \\rightarrow 0$ asymptotics}\\label{subsec:sigma:0}\n\\begin{Prop}$ \\underset{\\sigma \\to 0}{\\lim}\\, p_{\\sigma,1}=0$. \n\\end{Prop}\n\n\\begin{proof} This follows from the fact that the volume of $T(\\sigma) \\rightarrow 0$ as $\\sigma \\rightarrow 0$.\\end{proof}\n\n\\begin{Theorem}\\label{sig to 0} As $\\sigma \\rightarrow 0$,\n$\\sigma^{-2} p_{\\sigma,2}$ is bounded away from 0 and infinity.\n\\end{Theorem}\n\\begin{proof}\nConsider $g_{-\\sqrt{\\sigma}} T(c,\\sigma)$. Because the action of\n$g_t$ preserves $\\mu$ on $\\mathcal{H}$ it follows that the measure\nof surfaces with two saddle connections $T(c, \\sigma)$ is equal to\nthe measure of surfaces with two saddle connections in\n$g_{-\\sqrt{\\sigma}}T(c,\\sigma)$. It follows from Lemma\n\\ref{lemma:short:saddle:connections} and the remark following it that $$\\mu(\\{\\omega:\n\\Lambda_{\\omega}\\cap T(c,\\sigma)\\geq 2\\}) \\sim \\sqrt{\\sigma}^2\\sqrt{\\sigma}^2=\\sigma^2,$$ where $\\sim$ denotes proportionality.\n\\end{proof}\n\n\nThis result states that given a saddle connection appearing in a wedge the probability of having another one is roughly independent. In general this is false, as the next section shows the probability of having many saddle connections in a small wedge decays slowly.\n\n \n\n\n\\subsection{More on $\\sigma \\rightarrow 0, k \\rightarrow \\infty$ }\nFirst,\n\\begin{lemma} \\label{2k above} For any fixed $\\sigma>0$ we have $\\underset{k \\to \\infty}{\\limsup} \\, k^2 \\tilde{p}^{\\text{cyl}}_k(\\sigma)<\\infty$.\n\\end{lemma}\n\nThis is similar to the proof of Theorem \\ref{sig to 0}. We say a surface has an $\\epsilon$\\emph{-thin neck} if there exists a pair of saddle connections $\\mathbf v$\n and $\\mathbf w$ such that $|\\mathbf v| \\leq \\epsilon$ and $\\mathbf w$, $\\mathbf v$, $\\mathbf w$\n are adjacent saddle connections. See Figure~\\ref{thin neck} below. \n \\makefig{An $\\epsilon$-thin neck, i.e., $|\\mathbf v| < \\epsilon$.}{thin neck}{\\input{thin_neck.pstex_t}} \n\n\n If $\\mathbf v$, $\\mathbf w$ define a $\\frac{\\sigma}{3k}$ thin neck in $\\omega$ and \n $\\frac 1 3 \\leq |\\mathbf w| \\leq \\frac 2 3$ then $N_1^{\\arg (\\mathbf w)}(\\omega,\\sigma,0)\\geq k$.\n To see this notice that a saddle connection that goes once in the $\\mathbf w$ direction and $l$ times in the $\\mathbf v$\n direction has its associated vector in $\\Lambda_{\\omega}$ contained in $A_1^{\\arg(\\mathbf w)}(0,\\sigma)$. \n\\begin{lemma}\\label{k asym} For any fixed $\\sigma >0$ we have $ k^2 \\tilde{p}_k(\\sigma)$ is bounded away from 0 and $\\infty$.\n\\end{lemma}\n\\begin{lemma} \\label{sig asym} For any fixed $k>1$ we have $\\sigma^{-2} \\tilde{p}_k(\\sigma)$ is bounded away from $0$ and $\\infty$.\n\\end{lemma}\nThese two lemmas follow from the previous paragraph by noticing that the measure of surfaces with an $\\epsilon$-thin neck is proportional to $\\epsilon^2$ (see \\S \\ref{subsec:coord:meas}). Showing that it is bounded away from $\\infty$ follows from Theorem \\ref{sig to 0} and Lemma \\ref{2k above}.\n\n\\subsubsection{Non-existence of second moments}\\label{subsec:second:moment} Lemma~\\ref{k asym} has the following corollary.\n\n\\begin{Cor}\\label{cor:second:moment} For any $\\sigma >0$, $$\\sum_{k =0}^{\\infty} k^2 p_k(\\sigma) \\mbox{ diverges }.$$ That is, the limiting distribution $p_k(\\sigma)$ does not have finite second moment.\\end{Cor}\n\n\\medskip\n\\noindent The corollary follows from Lemma~\\ref{k asym} and the general lemma below:\n\n\\begin{lemma}\\label{lemma:prob} Let $X$ be a positive integer valued random variable, with $P(X = k) = p_k$. Suppose there is a $c >0$ so that $P(X \\geq k ) \\geq \\frac{c}{k^2}$. Then\n$$\\sum_{k =0}^{\\infty} k^2 p_k \\mbox{ diverges}.$$\n\\end{lemma}\n\n\\begin{proof} We are interested in calculating $E(X^2) = \\sum_{k=0}^{\\infty} k^2 p_k$. We can write\n$$E(X^2) = \\sum_{k=0}^{\\infty} P(X^2 \\geq k)$$ For $i^2 \\le k < (i+1)^2$, $P(X^2 \\geq k) = P(X \\geq i)$. Thus $$\\sum_{k=0}^{\\infty} P(X^2 \\geq k) = \\sum_{i=1}^{\\infty} 2i P(X \\geq i) \\geq \\sum_{i=1}^{\\infty} \\frac{c}{i}.$$\\end{proof}\n\n\n\n\\subsubsection{Glued-in tori} For the remaining estimates we consider gluing in a small torus.\n We say a surface has a \\emph{glued in\ntorus} with parameters $(a,b)$ and gluing slit $s$ there is a portion of the surface where the points travel as if they are in a torus with basis lengths $a,b$ except if they cross a saddle connection of length at most $s$. See Figure~\\ref{almost torus} below.\n\n \\makefig{The slit torus is glued to the rest of the surface along the saddle connection of length at most $s$.}{almost torus}{\\input{almost_torus.pstex_t}} \n\n\\begin{Prop}\n The measure of unit volume surfaces that have a glued in torus with\nparameters $(a,b)$ and gluing slit $s$ where $a, b \\in\n[\\sqrt{c},2\\sqrt{c}]$ and $s \\in [c,2c]$ is at least proportional to $c^{-4}$\nas $c $ goes to zero.\n\\end{Prop}\nThis follows from the main result in \\S \\ref{subsec:coord:meas}. If we glue in a torus with parameters comparable to $(\\sqrt{\\sigma}$,\n$\\sqrt{\\sigma})$ with gluing slit $\\sigma$ it has quadratic growth of\nsaddle connections or periodic cylinders. Because a torus is a\nlattice surface, the saddle connection directions are completely\nperiodic. If the length of the periodic cylinder is less than $t$\nthen the trajectory in the torus crosses the direction of the slit\nat most $c\\frac{t}{\\sqrt{\\sigma}}=ct\\sqrt{\\sigma}^{-1}$ times. \nSo some points do not leave the torus before\nclosing up. Therefore all saddle connection directions of the torus\nwith length less than $\\frac{\\sigma}{c}$ are cylinder directions for the\nsurface.\n\nIt follows that there exists $C>0$ such that for any $\\sigma>0$ small\nenough a surface that has a torus with gluing parameters\n$\\sqrt{\\sigma}$, $\\sqrt{\\sigma}$, not too small angle between these sides and gluing slit $\\sigma$ has at\nleast $C \\sigma^{-1}$ periodic directions whose cylinders have\nlength less than or equal to 1. It follows from the pigeonhole\nprinciple that almost half of these directions are separated by at\nmost $4\\pi C \\sigma^{-1}$.\n\n\n\n\\begin{lemma}\nFor any fixed $k>0$ we have\n$$\\underset{\\sigma \\to 0}{\\liminf}\\,\n\\sigma^{-4}\\, \\tilde{p}^{cyl}_k(\\sigma)>0.$$\n\\end{lemma}\n\n\n\\begin{proof}Glue in a torus of parameters comparable to $\\sqrt{\\sigma}$,$\\sqrt{\\sigma}$ with gluing slit $\\sigma$.\\end{proof}\n\\begin{lemma}For any fixed $\\sigma>0$ we have\n$$\\underset{k \\to \\infty}{\\liminf}\\, k^{4} \\tilde{p}^{cyl}_k>0.$$\n\\end{lemma}\n\\begin{proof}Glue in a torus of parameters comparable to $\\sqrt{2 \\sigma\nk}^{-1}$, $\\sqrt{2\\sigma k}^{-1}$ with gluing slit $(2\\sigma k)^{-1}$.\\end{proof}\n\n\n\\begin{Cor}\\label{SV not L2} Let $f: \\mathbb{R}^2 \\backslash \\{0\\} \\to \\mathbb{R}$ by $f(x)=1$ if $x \\in B(0,1) \\backslash \\{0\\}$ and 0 otherwise. Its Siegel-Veech transform $\\hat{f}$ is not in $L^{2}(\\mathcal H, \\mu)$. However $\\hat{f}$ is in $L^{2-\\epsilon}(\\mathcal H,\\mu)$ for any $\\epsilon>0$.\n\\end{Cor}\n\\medskip\n\n\\noindent This is an immediate consequence of Lemma \\ref{k asym}. Notice that $f$ is in $L^{\\infty}(\\mathbb{R}^2)$. The above corollary fails if we take the analogue of the Siegel-Veech transform for the directions of cylinders. That is, let \n$$\\tilde{f}(\\omega)= \\sum_{\\mathbf v \\in \\Lambda_{\\omega}^{cyl}} f(\\mathbf v) .$$ In this case the function is not in $L^{4}(\\mathcal H, \\mu)$. However, if we fix the minimal volume of cylinders we consider then by Lemma \\ref{cyl flow} this variant of the Siegel-Veech transform sends $L^{\\infty}(\\mathbb{R}^2\\backslash \\{0\\})$ to $L^{\\infty}(\\mathcal H,\\mu)$. The $L^{\\infty}$ norm may increase and that this increase can be bounded by the genus of the surfaces parametrized by $\\mathcal H$ and the lower bound on the volume of the cylinders. Thus while the different versions of the Siegel-Veech transform are all $L^1$ norm preserving on positive functions they have different behavior in $L^p$ in general.\n\n\\subsection{$\\sigma \\rightarrow \\infty$ asymptotics}\\label{subsec:sigma:infty}\n\\begin{lemma}\\label{cyl flow} Suppose $x$ is in a periodic cylinder of length $L$, area $a$ in direction $\\theta$. Then $x$ is not in a periodic cylinder of length less than $R$ and direction $\\theta'$ where $ \\theta' \\neq \\theta$ and $|\\theta'-\\theta|<\\min\\{\\frac 1 {2LRa}, \\frac{\\pi}{3}\\}$.\n\\end{lemma}\n\\begin{proof} Consider the periodic cylinder in the hypothesis of the lemma.\n It has a width vector $\\mathbf v$ where $|\\mathbf v|=\\frac{a}{L}$ and let $x \\in \\mathbf v$. If $x+L\\tan(\\epsilon)< \\frac a L$ then $F_{\\theta+\\epsilon}|_{\\mathbf v}(x)=x+ L \\tan(\\epsilon)$, where $F_{\\theta+\\epsilon}|_{\\mathbf v}$ denotes the induced map of $F_{\\theta+\\epsilon}$ on $\\mathbf v$. It follows that if $\\theta+\\epsilon$ is the direction of a periodic cylinder $x$ lies in we have $F_{\\theta+\\epsilon}|_{\\mathbf v}^k(x)=x$ and so $kL\\tan(\\epsilon)\\geq |\\mathbf v|.$ The length of this cylinder is at least $kL\\sec(\\epsilon)$. Therefore if $kL\\frac{|\\mathbf v|}{R}=\\frac{a}{RL}.$ Noticing that $\\epsilon< \\frac{\\pi}{3}$ implies $\\tan(\\epsilon)<2\\epsilon$ completes the lemma.\n\\end{proof}\n\n \n \\subsubsection{Absence of cylinder gaps}\\label{subsec:cylinder:gap}\nLet $\\Theta_{\\omega}^{*}(R)$ be the directions $\\theta$ such that the volume of the periodic cylinders in direction $\\theta$ with length than or equal to $R$ is at least $\\frac{3}{5}$. In the rest\n\\begin{lemma}\\label{farey} Let $\\theta \\in \\Theta_{\\omega}^{*}(L)$ then\n $$\\left(\\theta - \\frac{1}{20(4g-4)},\\theta+ \\frac{1}{20(4g-4)RL}\\right)\n\\cap \\Theta_{\\omega}^{*}(R)=\\theta.$$\n\\end{lemma}\n\\begin{proof} One can easily see that the number of \nsaddle connections in a given direction is at most $4g-4$.\n Therefore $\\theta \\in \\Theta_{\\omega}^{*}(L)$ \nthen at least half the points in the surface must lie in \ncylinders of area at least $\\frac{1}{10(4g-4)}$.\n If $\\phi \\in \\Theta_{\\omega}^{\\frac3 5}(R)$ then some points of the surface\n must lie in periodic cylinders in direction $\\phi$ and\n periodic cylinders of area at least $\\frac 1 {10(4g-4)}$\n in direction $\\theta.$\nThe result follows from Lemma \\ref{cyl flow}.\n\\end{proof}\nIn the square torus, periodic directions correspond to rational slopes. Notice that if $\\frac a L \\neq \\frac b R $ are rational numbers and $CL\\frac{C}{R^2}$. This implies that for any $C'>0$ a positive proportion of periodic directions of length less than $R$ are at least $\\frac{C'}{R^2}$ separated from the closest periodic direction of length less than $R$. The next theorem generalizes this fact for periodic cylinders of substantial area.\n\n\\begin{Theorem}\\label{p sig 0 lower}\nIf the cardinality of $\\Theta_{\\omega}^{*}(R)=\\{\\theta_1 \\leq \\theta_2 \\leq \\dots \\theta_N\\}$\n grows quadratically then \n $$\\underset{\\sigma \\to \\infty}{\\liminf} \\, \\underset{R \\to \\infty}{\\liminf} \\,\\sigma^2 \\frac{|\\{\\theta_i \\in \\Theta_{\\omega}^{*}(R): \\theta_{i+1}-\\theta_i \\geq \\frac{\\sigma}{R}\\}|}{|\\Theta_{\\omega}^{\\frac 3 5 }(R)|}>0.$$\n\\end{Theorem}\n\\begin{proof} By the fact that $\\omega$ has quadratic growth there exists $c>0$ such that $$\\frac{|\\Theta_{\\omega}^{*}(L)|}{|\\Theta_{\\omega}^{*}(R)|}\\geq c \\left(\\frac L R\\right)^2.$$ Lemma \\ref{farey} implies that if\n $\\theta \\in \\Theta_{\\omega}^{*}(L)$ then \nit has a gap of size at least $\\frac 1 {20(4g-4)RL}$ to the closest\ndirection in $\\Theta_{\\omega}^{*}(R)$. Let $L=\\frac {R}{\\sigma 20(4g-4)}$.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Billiards}\\label{sec:billiards} In this section we use Theorem~\\ref{theorem:wedge:limit} to obtain results about special trajectories for billiards, as discussed in \\S\\ref{subsubsec:billiards:barriers}. We follow closely the exposition in~\\cite{EMM, EMS}, particularly focusing on the examples studied in the latter paper.\n\n\n\\subsection{Rectangles with barriers}\\label{subsec:barrier} We recall notation from \\S\\ref{subsubsec:billiards:barriers}: given $\\alpha \\in \\bb R$, consider the polygon $P_{\\alpha}$ whose boundary is the boundary of the square $[0, 1] \\times [0, 1]$ together with a barrier given by the vertical segment $\\{1\/2\\} \\times [0, \\alpha]$. \n\nWe recall the `unfolding' procedure from~\\cite{EMS}: to obtain a translation surface $\\omega_{\\alpha}$ from $P_{\\alpha}$, take four copies of $P= P_{\\alpha}$ which are images of $P$\nunder reflection in the two coordinate axes and reflection in the origin. \n\nIdentifying the interior sides, we obtain a square of area 4 with two vertical double lines, corresponding to the interval $1\/2 \\times [0,\\alpha]$. Identify the top and bottom of the square, and the left and right sides. The glue the left side of the\nright line to the right side of the left line, and the right side of\nthe right line to the left side of the left line. See Figure~\\ref{unfold} below.\n\n\n\\makefig{Unfolding $P_{\\alpha}$ to $\\omega_{\\alpha}$.}{unfold}{\\input{developwall.pstex_t}} \n\n\nRescaling by $1\/4$, we obtain an area $1$ translation surface $\\omega_{\\alpha} \\in \\mathcal H(1,1)$, with the $2$ zeroes located at the endpoints of the vertical lines. A billiard trajectory $\\lambda$ on $P_{\\alpha}$ corresponds to a straight line on $\\omega_{\\alpha}$.\n\n\\subsection{Branched covers}\\label{subsec:branch} As mentioned in \\S\\ref{subsubsec:billiards:barriers}, the crucial property of the surface $\\omega_{\\alpha}$ is that it is a double (branched) cover of the torus. That is, there is a covering map (branched at the zeros) $\\pi: \\omega_{\\alpha} \\rightarrow \\bb C\/\\Lambda$ where $\\Lambda$ is a lattice in $\\bb C$ and that $\\omega_{\\alpha}$ is obtained by pulling back the form $dz$. \n\nThe set of all $\\omega \\in \\mathcal H(1, 1)$ satisfying this property is a closed, $SL(2, \\bb R)$-subvariety $\\mathcal M$~\\cite[Lemma 2.1]{EMS}. $\\mathcal M$ is a finite cover of $\\mathcal T = (SL(2, \\bb R) \\ltimes \\bb R^2)\/(SL(2, \\bb Z) \\ltimes \\bb Z^2)$ and the covering map commutes with the $SL(2,\\bb R)$-action~\\cite[Lemma 2.2]{EMS}. Recall that $\\mathcal T$ is the space of tori with two marked points (assuming that one marked point is always at the origin). The covering map $\\Pi: \\mathcal M \\rightarrow \\mathcal T$ is given by \n$$\\Pi(\\omega) = (\\Lambda, \\pi(z_1), \\pi(z_2)),$$ \n\\noindent where $\\Lambda \\subset \\bb C$ is the lattice so that $\\omega$ covers $\\bb C\/\\Lambda$ and $z_1, z_2$ are the zeros of $\\omega$. \n\nLet $\\alpha \\in \\bb R$ be irrational. For $\\omega_{\\alpha} \\in \\mathcal M$, let $\\mathcal M(\\alpha)$ denote the connected component of $\\mathcal M$ containing $\\omega_{\\alpha}$. Let $\\bar{\\mu}$ denote the pullback of the Haar probability measure on $\\mathcal T$. It is an ergodic $SL(2, \\bb R)$ invariant measure on $\\mathcal M(\\alpha)$. The circle limit measure for $\\omega_{\\alpha}$ is $\\bar{\\mu}$~\\cite[Lemma 2.4]{EMS}. Thus, we obtain the following corollary to Theorem~\\ref{theorem:wedge:limit}.\n\n\\begin{Cor}\\label{cor:wedge:billiard} For any irrational $\\alpha$,\n$$\\lim_{R \\rightarrow \\infty} \\lambda(\\theta: N^{\\theta}_R(\\omega_{\\alpha}, \\sigma, c) = k) = \\bar{\\mu}(\\omega \\in \\mathcal M(\\alpha): \\Lambda^{sc}_{\\omega} \\cap T(c, \\sigma) = k).$$\n\\end{Cor}\n\n\\subsubsection{Branched covers of lattice surfaces}\\label{subsubsec:branch:lattice} A similar calculation of circle limit measures was carried out for branched covers of lattice surfaces in~\\cite{EMM}. Thus, we can obtain a version of Corollary~\\ref{cor:wedge:billiard} for these surfaces as well. This will yield results for triangular billiards $P_n$ with angles\n $$ \\frac{n-2}{2n} \\pi, \\ \\frac{n-2}{2n} \\pi, \\ \\frac{4}{2n} \\pi\n ,$$ where $n \\ge 5$, $n$ odd. For details, see~\\cite[\\S9]{EMM}. \n\n\\subsection{Lattice translates}\\label{subsec:translate} To obtain a version of Theorem~\\ref{theorem:main:gap} for $\\omega \\in \\mathcal M$, we have to understand the set of holonomy vectors of saddle connections $\\Lambda^{sc}_{\\omega}$. Suppose $\\omega$ is a cover of $\\bb C\/\\Lambda$, with covering map $\\pi$. Note that under the projection to the torus, the saddle connections must connect either $\\pi(z_1) = 0$ or $\\pi(z_2)$ to themselves or to each other. That is, they must be primitive vectors in the lattice $\\Lambda$ or in the translate $\\Lambda + \\mathbf v$ where $\\mathbf v$ is a choice of vector connecting the two marked points on the torus (defined up to $\\Lambda$, so $\\Lambda + \\mathbf v$ is well-defined). Thus, if we have $\\Pi(\\omega) = (\\Lambda, \\mathbf v)$ (viewing $\\mathcal T$ as the space of marked tori, or equivalently, lattices and a choice of vector), we have $$\\Lambda^{sc}_{\\omega} = \\Lambda_{prim} \\cup ( \\Lambda_{prim} + \\mathbf v),$$ where $\\Lambda_{prim}$ denotes the set of primitive vectors in $\\Lambda$. \n\n$\\mathcal T$ is a fiber bundle over the modular surface $SL(2,\\bb R)\/SL(2, \\bb Z)$. It can be broken up into a Haar measure on $SL(2, \\bb R)\/SL(2,\\bb Z)$ together with Lebesgue measure on the torus fibers. While a lattice $\\Lambda$ will never have two points in $T(0, \\sigma)$ for $\\sigma < <1$, we can construct a positive measure set of pairs $(\\Lambda, \\mathbf v) \\in \\mathcal T$ so that $\\Lambda_{prim} \\cup (\\Lambda_{prim} + \\mathbf v)$ does intersect $T(\\sigma)$ (at least) twice for all $\\sigma >0$ by considering the lattices $\\Lambda$ so that $\\Lambda \\cap T(\\sigma) \\neq \\emptyset$ and $\\mathbf v \\in T(\\sigma)$ ($\\mathbf v \\notin \\Lambda$). Thus for all $\\alpha$ irrational, all $\\sigma >0$, we have\n$$\\bar{\\mu}(\\omega \\in \\mathcal M(\\alpha): \\Lambda^{sc}_{\\omega} \\cap T(\\sigma) \\geq 2) >0.$$\n\\noindent (In fact, we will obtain a set of measure proportional to $\\sigma^4$) (see~\\cite[Remark 2.3]{MS} for an explicit description of the distribution of gaps for the set $\\mathbf v + \\Lambda$). Thus, we obtain:\n\n\n\\begin{Cor}\\label{cor:main:gap} For all irrational $\\alpha$, \n\\begin{equation}\\label{eq:cor:gap}\\lim_{R \\rightarrow \\infty} R^2 \\gamma^{\\omega_{\\alpha}}(R) = 0.\n\\end{equation}\n\\noindent Moreover, for any $\\epsilon >0$, the proportion of gaps less than $\\epsilon\/R^2$ is positive. That is, writing $\\Theta^{\\omega_{\\alpha}}_R : = \\{0 \\le \\theta_1 \\le \\theta_2 \\le \\ldots \\le \\theta_n\\}$, we have\n\\begin{equation}\\label{eq:cor:proportion}\n\\lim_{R \\rightarrow \\infty} \\frac{ |\\{1 \\le i \\le \\tilde{N}(\\omega_{\\alpha}, R): (\\theta_{i+1} - \\theta_i) \\le \\epsilon\/R^2\\}|}{\\tilde{N}(\\omega_{\\alpha}, R)} >0\n\\end{equation}\n\\end{Cor}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Newton-Puiseux Theorem states that, if $f\\left(x,y\\right)$ is\nan analytic germ in two variables, then the solutions $y=\\varphi\\left(x\\right)$\nof the equation $f=0$ can be expanded as Puiseux series that are\nconvergent in a neighbourhood of the origin (see for example \\cite{brieskorn_knorrer;plane_algebraic_curve}).\nA multivariable version of this result in the real case states that,\nif $f\\left(x_{1},\\ldots,x_{m},y\\right)$ is a real analytic germ,\nthen, after a finite sequence of blow-ups with centre a real analytic\nmanifold, the solutions $y=\\varphi\\left(x_{1},\\ldots,x_{m}\\right)$\nof the equation $f=0$ are analytic in a neighbourhood of the origin\n(see for example \\cite[Theorem 4.1]{parusinski:preparation}). An\nequivalent formulation states that the solutions $y=\\varphi\\left(x_{1},\\ldots,x_{m}\\right)$\nin a neighbourhood of the origin are obtained, piecewise, as finite\ncompositions of analytic functions, taking $n^{\\text{th}}$ roots\nand quotients (see for example \\cite[Corollary 2.15]{dmm:exp} and\n\\cite[Theorem 1]{lr:prep}).\n\n\n\\paragraph{{}}\n\nHere we extend this result to functions belonging to a \\emph{generalised\nquasianalytic class} (see Definition \\ref{def:qa class}). Roughly,\na generalised quasianalytic class is a collection of algebras of continuous\nreal-valued functions together with an \\emph{injective} $\\mathbb{R}$-algebra\nmorphism $\\mathcal{T}$ which, given the germ at zero $f$ of a function\nin the collection, associates to $f$ a formal power series $\\mathcal{T}\\left(f\\right)$\nwith natural or real exponents. Given a generalised quasianalytic\nclass, we already have a local uniformisation result \\cite{rsw,martin_sanz_rolin_local_monomialization,rolin_servi:qeqa}\nwhich allows to parametrise the zero set of a function in the class.\nOur aim here is to refine this procedure, in the spirit of the elimination\nresult in \\cite{vdd:d}, in the following way: given a function $f\\left(x,y\\right)$\nin the class under consideration, we provide a uniformisation algorithm\nwhich ``respects'' the variable $y$ and hence allows to solve the\nequation $f=0$ with respect to $y$. \n\n\n\\paragraph{{}}\n\nExamples of generalised quasianalytic classes are the following (see\nRemark \\ref{rem: thm abcde}).\n\n\n\\paragraph{{}}\n\n\\noindent a) Let $M=\\left(M_{0},M_{1},\\ldots\\right)$ be an increasing\nsequence of positive real numbers (with $M_{0}\\geq1$) and $B\\subseteq\\mathbb{R}^{m}$\nbe a compact box. We assume that $M$ is strongly log-convex and we\nconsider the Denjoy-Carleman algebra of functions $\\mathcal{C}_{B}\\left(M\\right)$\ndefined in \\cite{rsw}. This is an algebra of functions $f:B\\to\\mathbb{R}$\nwhich each extend to a $\\mathcal{C}^{\\infty}$ function on some open\nneighbourhood $U\\supseteq B$ and whose derivatives satisfy a certain\ntype of bounds depending on $M$ (see \\cite[p. 751]{rsw}). The functions\nin $\\mathcal{C}_{B}\\left(M\\right)$ are not analytic in general, however,\nif $\\sum_{i\\in\\mathbb{N}}\\frac{M_{i}}{M_{i+1}}=\\infty$, then $\\mathcal{C}_{B}\\left(M\\right)$\nis \\emph{quasianalytic}, i.e. for every $x\\in B$, the algebra morphism\nwhich associates to $f\\in\\mathcal{C}_{B}\\left(M\\right)$ its (divergent)\nTaylor expansion at $x$ is injective. The quasianalytic Denjoy-Carleman\nclass $\\mathcal{C}\\left(M\\right)$ is the union of the collection\n$\\left\\{ \\mathcal{C}_{B}\\left(M\\right):\\ m\\in\\mathbb{N},\\ B\\subseteq\\mathbb{R}^{m}\\ \\text{compact\\ box}\\right\\} $. \n\n\n\\paragraph{{}}\n\nb) Let $H=\\left(H_{1},\\ldots,H_{r}\\right):\\left(0,\\varepsilon\\right)\\to\\mathbb{R}^{r}$\nbe a $\\mathcal{C}^{\\infty}$ solution of a system of first order singular\nanalytic differential equations of the form $x^{p+1}y'\\left(x\\right)=A\\left(x,y\\right)$,\nwhere $A$ is real analytic in a neighbourhood of $0\\in\\mathbb{R}^{p+1}$,\nsatisfying conditions a) and b) in \\cite[p. 413]{rss}, and $A\\left(0,0\\right)=0$.\nSuppose furthermore that $H$ admits an asymptotic expansion for $x\\rightarrow0^{+}$\nas in \\cite[2.2]{rss}. As in \\cite[Section 3]{rss}, we let $\\mathcal{A}_{H}$\nbe the smallest collections of real germs containing the germ at zero\nof the $H_{i}$ and closed under composition, monomial division and\ntaking implicit functions. A function $f$, defined on an open set\n$U\\subseteq\\mathbb{R}^{m}$, is said to be $\\mathcal{A}_{H}$-\\emph{analytic\n}if for every $a\\in U$ there exists a germ $\\varphi_{a}\\left(x\\right)\\in\\mathcal{A}_{H}$\nsuch that the germ of $f\\left(x\\right)$ at $a$ is equal to the germ\n$\\varphi_{a}\\left(x-a\\right)$. It is proven in \\cite{rss} that the\ncollection of all $\\mathcal{A}_{H}$-analytic functions forms a quasianalytic\nclass of $\\mathcal{C}^{\\infty}$ functions.\n\n\n\\paragraph{{}}\n\n\\noindent c) A (formal) generalised power series in $m$ variables\n$X=\\left(X_{1},\\ldots,X_{m}\\right)$ is a series $F\\left(X\\right)=\\sum_{\\alpha}c_{\\alpha}X^{\\alpha}$\nsuch that $\\alpha\\in[0,\\infty)^{m}$, $c_{\\alpha}\\in\\mathbb{R}$ and\nthere are well-ordered subsets $S_{1},\\ldots,S_{m}\\subseteq[0,\\infty)$\nsuch that the support of $F$ is contained in $S_{1}\\times\\ldots\\times S_{m}$\n(see \\cite{vdd:speiss:gen}). The series $F$ is convergent if there\nis a polyradius $r=\\left(r_{1},\\ldots,r_{m}\\right)\\in\\left(0,\\infty\\right)^{m}$\nsuch that $\\sum_{\\alpha}|c_{\\alpha}|r^{\\alpha}<\\infty$. A convergent\ngeneralised power series gives rise to a real-valued function $F\\left(x\\right)=\\sum c_{\\alpha}x^{\\alpha}\\in\\mathbb{R}\\left\\{ x^{*}\\right\\} _{r}$,\nwhich is continuous on $[0,r_{1})\\times\\ldots\\times[0,r_{m})$ and\nanalytic on the interior of its domain. We denote by $\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket $\nthe algebra of all formal generalised power series and consider the\nalgebra $\\mathbb{R}\\left\\{ x^{*}\\right\\} =\\bigcup_{r\\in\\left(0,\\infty\\right)^{m}}\\mathbb{R}\\left\\{ x^{*}\\right\\} _{r}$\nof all convergent generalised power series. Examples of convergent\ngeneralised power series are the function $\\zeta\\left(-\\log x\\right)=\\sum_{n=1}^{\\infty}x^{\\log n}$\n(where $\\zeta$ is the Riemann zeta function) and the solution $f\\left(x\\right)=\\sum_{n,i=0}^{\\infty}\\frac{1}{2^{i}}x^{2+n-\\frac{1}{2^{i}}}$\nof the functional equation $\\left(1-x\\right)f\\left(x\\right)=x+\\frac{1}{2}x\\left(1-\\sqrt{x}\\right)f\\left(\\sqrt{x}\\right)$.\n\n\n\\paragraph{{}}\n\n\\noindent d) For $R=\\left(R_{1},\\ldots,R_{m}\\right)\\in\\left(0,\\infty\\right)^{m}$\na polyradius, we consider the algebra $\\mathcal{G}\\left(R\\right)$\nof functions defined in \\cite[Definition 2.20]{vdd:speiss:multisum}\nby means of sums of multisummable formal series in the real direction.\nIts elements are $\\mathcal{C}^{\\infty}$ functions defined on $\\left[0,R_{1}\\right]\\times\\ldots\\times\\left[0,R_{m}\\right]$\nand their derivatives satisfy a Gevrey condition. By a known result\nin multisummability theory, these algebras satisfy the following quasianalyticity\ncondition: the morphism, which associates to the germ at zero of a\nfunction in $\\mathcal{G}\\left(R\\right)$ its (divergent) Taylor expansion\nat the origin, is injective (see \\cite[Proposition 2.18]{vdd:speiss:multisum}).\nWe let $\\mathcal{G}$ be the union of the collection $\\left\\{ \\mathcal{G}\\left(R\\right):\\ m\\in\\mathbb{N},\\ R\\in\\left(0,\\infty\\right)^{m}\\right\\} $.\nThis collection contains the function $\\psi\\left(x\\right)$ appearing\nin Binet's second formula, i.e. such that $\\log\\Gamma\\left(x\\right)=\\left(x-\\frac{1}{2}\\right)\\log\\left(x\\right)+\\frac{1}{2}\\log\\left(2\\pi\\right)+\\psi\\left(\\frac{1}{x}\\right)$,\nwhere $\\Gamma$ is Euler's Gamma function (see \\cite[Example 8.1]{vdd:speiss:multisum}). \n\n\n\\paragraph{{}}\n\n\\noindent e) For $r\\in\\left(0,\\infty\\right)^{m+n}$ a polyradius,\nwe consider the algebra $\\mathcal{Q}_{m,n,r}$ defined in \\cite[Definition 7.1]{krs}.\nIts elements are continuous real-valued functions which have a holomorphic\nextension to some ``quadratic domain'' $U\\subseteq\\mathbf{L}^{m+n}$,\nwhere $\\mathbf{L}$ is the Riemann surface of the logarithm. One can\ndefine a morphism $T$ which associates to the germ $f$ of a function\nin $\\mathcal{Q}_{m,n,r}$ an \\emph{asymptotic expansion} $T\\left(f\\right)\\in\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket $.\nIt is shown in \\cite[Proposition 2.8]{krs}, using results of Ilyashenko's\nin \\cite{ilyashenko:dulac}, that the morphism $T$ is injective (quasianalyticity).\nWe let $\\mathcal{Q}$ be the collection $\\left\\{ \\mathcal{Q}_{m,n,r}:\\ m,n\\in\\mathbb{N},\\ r\\in\\left(0,\\infty\\right)^{m+n}\\right\\} $.\nThe motivation for looking at this type of algebras is that they contain\nthe Dulac transition maps of real analytic planar vector fields in\na neighbourhood of hyperbolic non-resonant singular points. \n\n\n\\paragraph{{}}\n\nBefore stating our main result, we need to give a definition. \n\\begin{defn}\n\\label{def: cell}Let $\\mathcal{A}$ be a collection of real-valued\nfunctions. An $\\mathcal{A}$\\emph{-term} is defined inductively as\nfollows. An $\\mathcal{A}$-term of depth zero is an element of $\\mathcal{A}$.\nLet $x=\\left(x_{1},\\ldots,x_{m}\\right)$. A function $f\\left(x\\right)$\nis an $\\mathcal{A}$-term of depth $\\leq k$ if there exist $m\\in\\mathbb{N},\\ g\\in\\mathcal{A}$\nand $\\mathcal{A}$-terms $t_{1}\\left(x\\right),\\ldots,t_{m}\\left(x\\right)$\nof depth $\\leq k-1$ such that $\\text{Im}\\left(t_{1}\\right)\\times\\ldots\\times\\text{Im}\\left(t_{m}\\right)\\subseteq\\text{dom}\\left(g\\right)$\nand $f\\left(x\\right)=g\\left(t_{1}\\left(x\\right),\\ldots,t_{m}\\left(x\\right)\\right)$.\n\nA connected set $C\\subseteq\\mathbb{R}^{m}$ is an $\\mathcal{A}$\\emph{-base}\nif there are a polyradius $r\\in\\left(0,\\infty\\right)^{m}$ and $\\mathcal{A}$-terms\n$t_{0},t_{1},\\ldots,t_{q}$ defined on $(0,r_{1})\\times\\ldots\\times(0,r_{m})$,\nsuch that\n\\[\nC=\\left\\{ x\\in(0,r_{1})\\times\\ldots\\times(0,r_{m}):\\ t_{0}\\left(x\\right)=0,\\ t_{1}\\left(x\\right)>0,\\ \\ldots,t_{q}\\left(x\\right)>0\\right\\} .\n\\]\nA set $D\\subseteq\\mathbb{R}^{m+1}$ is an $\\mathcal{A}$\\emph{-cell\n}if there are an $\\mathcal{A}$-base $C\\subseteq\\mathbb{R}^{m}$ and\nterms $t_{1}\\left(x\\right),t_{2}\\left(x\\right)$ in $m$ variables\nsuch that $D$ is of either of the following forms:\n\\begin{align*}\n\\left\\{ \\left(x,y\\right):\\ x\\in C,\\ y=t_{1}\\left(x\\right)\\right\\} , & \\ \\left\\{ \\left(x,y\\right):\\ x\\in C,\\ t_{1}\\left(x\\right)0\\\\\n0 & \\text{if}\\ x\\leq0\n\\end{cases}$ (for all $p\\in\\mathbb{N}$). \n\nWe can now state our main result.\n\\begin{namedthm}\n{Main Theorem}\\label{thm abcd}Let $\\mathcal{C}$ a generalised quasianalytic\nclass, as in Definition \\ref{def:qa class}. Let $\\mathcal{A}=\\mathcal{C}\\cup\\left\\{ \\left(\\cdot\\right)^{-1}\\right\\} \\cup\\left\\{ \\sqrt[p]{\\cdot}:\\ p\\in\\mathbb{N}\\right\\} $\nand $x=\\left(x_{1},\\ldots,x_{m}\\right)$. Let $y$ be a single variable\nand let $f\\left(x,y\\right)\\in\\mathcal{C}$. Then there exist a neighbourhood\n$W\\subseteq\\mathbb{R}^{m+1}$ of the origin and an $\\mathcal{A}$-cell\ndecomposition of $W\\cap\\text{dom}\\left(f\\right)$ which is compatible\nwith the set $\\left\\{ \\left(x,y\\right)\\in W\\cap\\text{dom}\\left(f\\right):\\ f\\left(x,y\\right)=0\\right\\} $.\n\\end{namedthm}\nThe Main Theorem immediately implies that the solutions of the equation\n$f\\left(x,y\\right)=0$ have the form $\\varphi:C\\to\\mathbb{R}$, where\n$C\\subseteq\\mathbb{R}^{m}$ is an $\\mathcal{A}$-base and $\\varphi\\left(x\\right)$\nis an $\\mathcal{A}$-term. \n\n\n\\paragraph{{}}\n\nWe now briefly illustrate the strategy of proof. In analogy with the\nreal analytic case, we define a class of blow-up transformations adapted\nto the functions under consideration. We show that, after a finite\nsequence of such transformations, the germ at zero of $f$ is normal\ncrossing.\n\nWe stress that the monomialisation algorithm we exhibit here differs\nfrom the ones in \\cite{rsw,martin_sanz_rolin_local_monomialization,bm_semi_subanalytic}.\nIn fact, the transformations we use \\emph{respect} the variable $y$\nin the following way: if $\\rho:\\mathbb{R}^{m+1}\\ni\\left(x',y'\\right)\\mapsto\\left(x,y\\right)\\in\\mathbb{R}^{m+1}$\nis one of such transformations and the Main Theorem holds for $f\\circ\\rho\\left(x',y'\\right)$,\nthen it also holds for $f\\left(x,y\\right)$. Moreover, such transformations\nare bijective outside a set of small dimension and the components\nof the inverse map, when defined, are $\\mathcal{A}$-terms. \n\nIt is worth pointing out that our algorithm does not use the Weierstrass\nPreparation Theorem, since this theorem does not always hold in generalised\nquasianalytic classes (see for example \\cite{parusinski_rolin:weierstrass_quasianalytic}).\n\n\n\\paragraph{{}}\n\nThe desingularisation procedure which allows to reduce to the case\nwhen $f$ is normal crossing exploits the fundamental property of\nquasianalyticity, which allows to deduce the wanted result for $f$\nfrom a formal monomialisation algorithm for the series $\\mathcal{T}\\left(f\\right)$.\n\n\n\\paragraph{{}}\n\nThe Main Theorem could also be deduced from a general quantifier elimination\nresult in \\cite{rolin_servi:qeqa}. However, the solving process described\nin \\cite{rolin_servi:qeqa} is not algorithmic, since it uses a highly\nnonconstructive result, namely an o-minimal Preparation Theorem in\n\\cite{vdd:speiss:preparation_theorems}. Here instead we deduce the\nexplicit form of the solutions of $f=0$ solely from the analysis\nof the Newton polyhedron of $\\mathcal{T}\\left(f\\right)$. \n\nAlthough all known generalised quasianalytic classes generate o-minimal\nstructures (see \\cite{vdd:tame} for the definition and basic properties\nof o-minimal structures), the proof of our main result does not use\no-minimality.\n\n\n\\section{Generalised quasianalytic classes}\n\nIn this section we establish our setting.\n\nWe recall the definition and main properties of generalised power\nseries (see \\cite{vdd:speiss:gen} for more details). \n\nLet $m\\in\\mathbb{N}$. A set $S\\subset[0,\\infty)^{m}$ is called \\emph{good}\nif $S$ is contained in a cartesian product $S_{1}\\times\\ldots\\times S_{m}$\nof well ordered subsets of $[0,\\infty)$. If $S$ is a good set, define\n$S_{\\mathrm{min}}$ as the set of minimal elements of $S$ with respect\nto the following order: let $s=\\left(s_{1},\\ldots,s_{m}\\right),\\ s'=\\left(s_{1}',\\ldots,s_{m}'\\right)\\in S;$\nthen $s\\leq s'$ iff $s_{i}\\leq s_{i}'$ for all $i=1,\\ldots,m$.\nBy \\cite[Lemma 4.2]{vdd:speiss:gen}, $S_{\\mathrm{min}}$ is finite. \n\nA \\emph{formal generalised power series }has the form\n\\[\nF(X)=\\sum_{\\alpha}c_{\\alpha}X^{\\alpha},\n\\]\n where $\\alpha=(\\alpha_{1},\\ldots,\\alpha_{m})\\in[0,\\infty)^{m},\\ c_{\\alpha}\\in\\mathbb{R}$\nand $X^{\\alpha}$ denotes the formal monomial $X_{1}^{\\alpha_{1}}\\cdot\\ldots\\cdot X_{m}^{\\alpha_{m}}$,\nand the \\emph{support of }$F$ $\\text{Supp}\\left(F\\right):=\\left\\{ \\alpha:\\ c_{\\alpha}\\not=0\\right\\} $\nis a good set. These series are added the usual way and form an $\\mathbb{R}$-algebra\ndenoted by $\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket $. \n\nLet $\\mathcal{G}\\subseteq\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket $\nbe a family of series such that the \\emph{total support} $\\text{Supp}\\left(\\mathcal{G}\\right):=\\bigcup_{F\\in\\mathcal{G}}\\text{Supp}\\left(F\\right)$\nis a good set. Then $\\text{Supp}\\left(\\mathcal{G}\\right)_{\\text{min}}$\nis finite and we denote by $\\mathcal{G}_{\\text{min}}:=\\left\\{ X^{\\alpha}:\\ \\alpha\\in\\text{Supp}\\left(\\mathcal{G}\\right)_{\\text{min}}\\right\\} $\nthe \\emph{set of minimal monomials} of $\\mathcal{G}$.\n\n\n\\paragraph{{}}\n\nLet $m,n\\in\\mathbb{N}$ and $(X,Y)=(X_{1},\\ldots,X_{m},Y_{1},\\ldots,Y_{n})$.\nWe define $\\mathbb{R}\\llbracket X^{*},Y\\rrbracket$ as the subring\nof $\\mathbb{R}\\llbracket(X,Y)^{*}\\rrbracket$ consisting of those\nseries $F$ such that $\\text{Supp}(F)\\subset[0,\\infty)^{m}\\times\\mathbb{N}^{n}$.\nSince $\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket \\subseteq\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket \\left\\llbracket Y\\right\\rrbracket $,\nwe say that the variables $X$ are \\emph{generalised} and that the\nvariables $Y$ are \\emph{standard}.\n\\begin{void}\n\\label{vuoto: functions}For every $m,n\\in\\mathbb{N}$ and polyradius\n$r=\\left(s_{1},\\ldots,s_{m},t_{1},\\ldots,t_{n}\\right)\\in\\left(0,\\infty\\right)^{m+n}$,\nwe let $\\mathcal{C}_{m,n,r}$ be an algebra of real functions, which\nare defined and continuous on the set\n\\[\nI_{m,n,r}:=[0,s_{1})\\times\\ldots\\times[0,s_{m})\\times\\left(-t_{1},t_{1}\\right)\\times\\ldots\\times\\left(-t_{n},t_{n}\\right),\n\\]\nand $\\mathcal{C}^{1}$ on $\\mathring{I}_{m,n,r}$. We denote by $x=\\left(x_{1,}\\ldots,x_{m}\\right)$\nthe \\emph{generalised} variables and by $y=\\left(y_{1},\\ldots,y_{n}\\right)$\nthe \\emph{standard} variables. We require that the algebras $\\mathcal{C}_{m,n,r}$\nsatisfy the following list of conditions:\\end{void}\n\\begin{itemize}\n\\item The coordinate functions of $\\mathbb{R}^{m+n}$ are in $\\mathcal{C}_{m,n,r}$. \n\\item If $r'\\leq r$ (i.e. if $s_{i}'\\leq s_{i}$ for all $i=1,\\ldots,m$\nand $t_{j}'\\leq t_{j}$ for all $j=1,\\ldots,n$) and $f\\in\\mathcal{C}_{m,n,r}$,\nthen $f\\restriction I_{m,n,r'}\\in\\mathcal{C}_{m,n,r'}$.\n\\item If $f\\in\\mathcal{C}_{m,n,r}$ then there exists $r'>r$ and $g\\in\\mathcal{C}_{m,n,r'}$\nsuch that $g\\restriction I_{m,n,r}=f$.\n\\item Let $k,l\\in\\mathbb{N}$, $s_{1}',\\ldots,s_{k}',t_{1}',\\ldots,t_{l}'\\in\\left(0,\\infty\\right)$\nand $r'=\\left(s_{1},\\ldots,s_{m},s_{1}',\\ldots,s_{k}',t_{1},\\ldots,t_{n},t_{1}',\\ldots,t_{l}'\\right)$.\nThen $\\mathcal{C}_{m,n,r}\\subset\\mathcal{C}_{m+k,n+l,r'}$ in the\nsense that if $f\\in\\mathcal{C}_{m,n,r}$ then the function\n\\[\n\\xyC{0mm}\\xyL{0mm}\\xymatrix{F\\colon & I_{m+k,n+l,r'}\\ar[rrrr] & \\ & \\ & \\ & \\mathbb{R}\\\\\n & \\left(x_{1},\\ldots,x_{m},x_{1}',\\ldots,x_{k}',y_{1},\\ldots,y_{n},y_{1}',\\ldots,y_{l}'\\right)\\ar@{|->}[rrrr] & & & & f\\left(x_{1},\\ldots,x_{m},y_{1},\\ldots,y_{n}\\right)\n}\n\\]\nis in $\\mathcal{C}_{m+k,n+l,r'}$.\n\\item $\\mathcal{C}_{m,n,r}\\subset\\mathcal{C}_{m+n,0,r}$, in the sense that\nif $f\\in\\mathcal{C}_{m,n,r}$ then $f\\restriction I_{m+n,0,r}\\in\\mathcal{C}_{m+n,0,r}$. \\end{itemize}\n\\begin{defn}\n\\label{def: quasi-analyticity}We denote by $\\mathcal{C}_{m,n}$ the\nalgebra of germs at the origin of the elements of $\\mathcal{C}_{m,n,r}$,\nfor $r$ a polyradius in $\\left(0,\\infty\\right)^{m+n}$. We say that\n$\\left\\{ \\mathcal{C}_{m,n}:\\ m,n\\in\\mathbb{N}\\right\\} $ is a collection\nof \\emph{quasianalytic algebras of germs} if, for all $m,n\\in\\mathbb{N}$,\nthere exists an \\textbf{injective} $\\mathbb{R}$-algebra morphism\n\\[\n\\mathcal{T}_{m,n}:\\mathcal{C}_{m,n}\\to\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket ,\n\\]\nwhere $X=\\left(X_{1},\\ldots,X_{m}\\right)=\\mathcal{T}\\left(x\\right),\\ Y=\\left(Y_{1},\\ldots,Y_{n}\\right)=\\mathcal{T}\\left(y\\right)$.\nMoreover, for all $m'\\geq m,\\ n'\\geq n$ we require that the morphism\n$\\mathcal{T}_{m',n'}$ extend $\\mathcal{T}_{m,n}$, hence, from now\non we will write $\\mathcal{T}$ for $\\mathcal{T}_{m,n}$.\n\nA number $\\alpha\\in[0,\\infty)$ is an \\emph{admissible exponent} if\nthere are $m,n\\in\\mathbb{N},$ $f\\in\\mathcal{C}_{m,n},\\ \\beta\\in\\text{Supp}\\left(\\mathcal{T}\\left(f\\right)\\right)\\subset\\mathbb{R}^{m}\\times\\mathbb{N}^{n}$\nsuch that $\\alpha$ is a component of $\\beta$. If $\\mathbb{A}$ is\nthe set of all admissible exponents and $\\mathbb{A}\\not=\\mathbb{N}$,\nthen we let $\\mathbb{K}$ be the set of nonnegative elements of the\nfield generated by $\\mathbb{A}$. Otherwise, we set $\\mathbb{K}=\\mathbb{A}=\\mathbb{N}$.\n\\end{defn}\nWe require the collection $\\left\\{ \\mathcal{C}_{m,n}:\\ m,n\\in\\mathbb{N}\\right\\} $\nto be closed under certain operations, which we now define.\n\\begin{defn}\n\\label{def:elem transf}Let $m,n\\in\\mathbb{N},\\ \\left(x,y\\right)=\\left(x_{1},\\ldots,x_{m},y_{1},\\ldots,y_{n}\\right)$.\nFor $m',n'\\in\\mathbb{N}$ with $m'+n'=m+n$, we set $\\left(x',y'\\right)=\\left(x_{1}',\\ldots,x_{m'}',y_{1}',\\ldots,y_{n'}'\\right)$.\nLet $r,r'$ be polyradii in $\\mathbb{R}^{m+n}$. An \\emph{elementary\ntransformation} is a map $I_{m',n',r'}\\ni\\left(x',y'\\right)\\mapsto\\left(x,y\\right)\\in I_{m,n,r}$\nof either of the following forms.\n\\begin{itemize}\n\\item A \\emph{ramification}: let $m=m',n=n'$, $\\gamma\\in\\mathbb{K}^{>0}$\nand $1\\leq i\\leq m$, and set\n\\begin{align*}\nr_{i}^{\\gamma} & \\left(x',y'\\right)=\\left(x,y\\right),\\ \\ \\ \\mathrm{where}\\ \\begin{cases}\nx_{k}=x_{k}' & 1\\leq k\\leq m,\\ k\\not=i\\\\\nx_{i}=x{}_{i}'^{\\gamma}\\\\\ny_{k}=y_{k} & 1\\leq k\\leq n\n\\end{cases}.\n\\end{align*}\n\n\\item A \\emph{Tschirnhausen translation}: let $m=m',n=n'$ and $H\\in\\mathcal{C}_{m,n-1,s}$\n(where $s\\in\\left(0,\\infty\\right)^{m+n-1}$ is a polyradius), with\n$H\\left(0\\right)=0$, and set\n\\[\n\\tau_{H}\\left(x',y'\\right)=\\left(x,y\\right),\\ \\ \\ \\mathrm{where}\\ \\begin{cases}\nx_{k}=x{}_{k}' & 1\\leq k\\leq m\\\\\ny_{n}=y_{n}'+H\\left(x',y_{1}',\\ldots,y_{n-1}'\\right)\\\\\ny_{k}=y_{k}' & 1\\leq k\\leq n-1\n\\end{cases}.\n\\]\n\n\\item A \\emph{linear transformation}: let $m=m',n=n'$, $1\\leq i\\leq n\\mathrm{\\ and\\ }c=\\left(c_{1},\\ldots,c_{i-1}\\right)\\in\\mathbb{R}^{i-1}$,\nand set\n\\[\nL_{i,c}\\left(x',y'\\right)=\\left(x,y\\right),\\ \\ \\ \\mathrm{where}\\ \\begin{cases}\nx_{k}=x_{k}' & 1\\leq k\\leq m\\\\\ny_{k}=y_{k}' & i\\leq k\\leq n\\\\\ny_{k}=y_{k}'+c_{k}y_{i}' & 1\\leq k1$ and $m_{i}=m_{i-1}'$\nfor all $i=1,\\ldots,k$, then we say that $\\rho:=\\nu_{1}\\circ\\ldots\\circ\\nu_{k}$\nis an \\emph{admissible transformation}.\n\nAn \\emph{elementary family }is either of the following collections\nof elementary transformations: $\\left\\{ r_{i}^{\\gamma}\\right\\} $\n$\\text{(for\\ some}\\ 1\\leq i\\leq m\\text{)},$ $\\left\\{ \\sigma_{m+i}^{+},\\sigma_{m+i}^{-}\\right\\} $\n$\\text{(for\\ some}\\ 1\\leq i\\leq n\\text{)},$ $\\left\\{ \\tau_{H}\\right\\} ,$\n$\\left\\{ L_{i,c}\\right\\} $ $\\text{(for\\ some}\\ 1\\leq i\\leq n\\text{)},$\n$\\left\\{ \\pi_{i,j}^{\\lambda}:\\ \\lambda\\in\\left[0,\\infty\\right]\\right\\} $\n$\\text{(for\\ some}\\ 1\\leq i,j\\leq m\\text{)},$ or $\\left\\{ \\pi_{m+i,j}^{\\lambda}:\\ \\lambda\\in\\mathbb{R}\\cup\\left\\{ \\pm\\infty\\right\\} \\right\\} $\n$\\text{(for\\ some}\\ 1\\leq i\\leq n,\\ 1\\leq j\\leq m\\text{)}$. An \\emph{admissible\nfamily} is defined inductively. An admissible family of length $1$\nis an elementary family. An admissible family $\\mathcal{F}$ of length\n$\\leq q$ is obtained from an elementary family $\\mathcal{F}_{0}$\nin the following way: for all $\\nu\\in\\mathcal{F}_{0}$, let $\\mathcal{F}_{\\nu}$\nbe an admissible family of length $\\leq q-1$ such that $\\forall\\rho'\\in\\mathcal{F}_{\\nu},\\ \\nu\\circ\\rho'$\nis an admissible transformation and define $\\mathcal{F}=\\left\\{ \\nu\\circ\\rho':\\ \\nu\\in\\mathcal{F}_{0},\\ \\rho'\\in\\mathcal{F}_{\\nu}\\right\\} $.\n\nFinally, we say that a series $F\\in\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nhas a certain property $P$ \\emph{after admissible family} if there\nexists ad admissible family $\\mathcal{F}$ such that for every $\\rho\\in\\mathcal{F}$\nthe series $F\\circ\\rho\\left(X',Y'\\right)$ has the property $P$.\nThe same notation extends to elements of $\\mathcal{C}$.\n\\end{defn}\nWe fix a generalised quasianalytic class $\\mathcal{C}$ and we let\n$\\widehat{\\mathcal{C}}_{m,n}$ be the image of $\\mathcal{C}_{m,n}$\nunder the morphism $\\mathcal{T}$ and $\\widehat{\\mathcal{C}}=\\bigcup\\widehat{\\mathcal{C}}_{m,n}$.\nIt follows from the conditions in \\ref{emp:properties of the morph}\nthat, if $\\rho:I_{m',n',r'}\\ni\\left(x',y'\\right)\\mapsto\\left(x,y\\right)\\in I_{m,n,r}$\nis an admissible transformation and $F\\left(X,Y\\right)\\in\\widehat{\\mathcal{C}}_{m,n}$,\nthen $F\\left(X',Y'\\right)\\in\\widehat{\\mathcal{C}}_{m',n'}$. \n\nMoreover, it is easy to verify that if $\\mathcal{G}\\subseteq\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nis a collection with good total support, then the collection $\\left\\{ F\\circ\\rho:\\ F\\in\\mathcal{G}\\right\\} $\nhas good total support. For example, let $F\\in\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nand $H\\in\\mathbb{R}\\left\\llbracket X^{*},Y_{1},\\ldots,Y_{n-1}\\right\\rrbracket $;\nsuppose $\\mathrm{Supp}\\left(F\\right)\\subseteq S_{1}\\times\\ldots\\times S_{m}\\times\\mathbb{N}^{n}$\nand $\\mathrm{Supp}\\left(H\\right)\\subseteq S'_{1}\\times\\ldots\\times S'_{m}\\times\\mathbb{N}^{n-1}$,\nwhere $S_{i},S'_{i}\\subset[0,\\infty)$ are well ordered sets. Then\nwe have $\\mathrm{Supp}\\left(F\\circ L_{i,c}\\right)\\subseteq S_{1}\\times\\ldots\\times S_{m}\\times\\mathbb{N}^{n}$\nand $\\mathrm{Supp}\\left(F\\circ\\tau_{H}\\right)\\subseteq\\tilde{S}_{1}\\times\\ldots\\times\\tilde{S}_{m}\\times\\mathbb{N}^{n}$,\nwith $\\tilde{S}_{k}=\\left\\{ a+lb:\\ a\\in S_{k},\\ b\\in S'_{k},\\ l\\in\\mathbb{N}\\right\\} $.\nMoreover, $\\mathrm{Supp}\\left(F\\circ r_{i}^{\\gamma}\\right)\\subseteq\\tilde{S}_{1}\\times\\ldots\\times\\tilde{S}_{m}\\times\\mathbb{N}^{n}$,\nwith $\\tilde{S}_{i}=\\left\\{ \\gamma a:\\ a\\in S_{i}\\right\\} $ and $\\tilde{S}_{k}=S_{k}$\nfor $k\\not=i$. Finally, for $1\\leq i,j\\leq m$ with $i\\not=j$, we\nhave $\\mathrm{Supp}\\left(F\\circ\\pi_{i,j}^{0}\\right)\\subseteq\\tilde{S}_{1}\\times\\ldots\\times\\tilde{S}_{m}\\times\\mathbb{N}^{n}$,\nwith $\\tilde{S}_{j}=\\left\\{ a+b:\\ a\\in S_{j},\\ b\\in S_{i}\\right\\} $\nand $\\tilde{S}_{k}=S_{k}$ for $k\\not=j$. The argument for the other\ntypes of blow-up transformation and for reflections is similar.\n\\begin{void}\n\\label{vuoto: normal}A series $F\\in\\widehat{\\mathcal{C}}_{m,n}$\nis \\emph{normal }if there are $\\alpha\\in[0,\\infty)^{m},\\ \\beta\\in\\mathbb{N}^{n}$\nand a unit $U\\in\\left(\\widehat{\\mathcal{C}}_{m,n}\\right)^{\\times}$\nsuch that $F\\left(X,Y\\right)=X^{\\alpha}Y^{\\beta}U\\left(X,Y\\right)$. \\end{void}\n\\begin{notation}\nThroughout this section, we let $m,n\\in\\mathbb{N},\\ \\left(x,y\\right)=\\left(x_{1},\\ldots,x_{m},y_{1},\\ldots,y_{n}\\right)$\nand $z$ be a single variable. We let $\\mathcal{C}_{m,n,1}$ be either\n$\\mathcal{C}_{m,n+1}$ (i.e. $z$ is considered as a standard variable)\nor $\\mathcal{C}_{m+1,n}$ (i.e. $z$ is considered as a generalised\nvariable). The same convention applies to the formal variables $X,Y,Z$\nand to $\\widehat{\\mathcal{C}}$.\n\\end{notation}\nLet $f\\left(x,y,z\\right)\\in\\mathcal{C}_{m,n,1}$. Our first aim is\nto show that, after a family of admissible transformations \\textquotedblleft{}respecting\\textquotedblright{}\\emph{\n}$Z$, the series $\\mathcal{T}\\left(f\\right)\\left(X,Y,Z\\right)$ is\nnormal. This motivates the next definition.\n\\begin{defn}\n\\label{def: vertical}Let $\\nu:I_{m',n'+1,r'}\\ni\\left(x',y',z'\\right)\\mapsto\\left(x,y,z\\right)\\in I_{m,n+1,r}$\nbe an elementary transformation. Let $\\nu_{0},r'_{0},r_{0}$ denote\nthe first $m+n$ components of $\\nu,r',r$ respectively. We say that\n$\\nu$ \\emph{respects} \\emph{the variable} $z$ if $\\nu_{0}$ does\nnot depend on $z'$. Hence $\\nu_{0}:I_{m',n',r'_{0}}\\ni\\left(x',y'\\right)\\mapsto\\left(x,y\\right)\\in I_{m,n,r_{0}}$\nis an elementary transformation. Analogously, we extend this definition\nto the case when $z'$ and$\\slash$or $z$ are generalised variables\nby requiring that the components of $\\nu$ which correspond to the\nvariables $\\left(x,y\\right)$ depend only on $\\left(x',y'\\right)$\nand not on $z'$.\\end{defn}\n\\begin{lem}\n\\label{lem: singular set}Suppose that $\\nu$ respects $z$, as in\nthe above definition. Then there exists a set $S\\subseteq I_{m',n',r'_{0}}$\n(which is either empty or the zeroset of some variable) such that\nthe maps $\\nu\\restriction I_{m',n'+1,r'}\\setminus\\left(S\\times\\mathbb{R}\\right)$\nand $\\nu_{0}\\restriction I_{m',n',r'_{0}}\\setminus S$ are bijections\nonto their image and for all $\\left(x',y'\\right)\\in I_{m',n',r'_{0}}\\setminus S$\nthe map $z'\\mapsto z=\\nu_{m+n+1}\\left(x',y',z'\\right)$ is a monotonic\nbijection onto its image. Moreover, the components of the inverse\nmaps $\\left(x,y\\right)\\mapsto\\left(x',y'\\right)$ and $\\left(x',y',z\\right)\\mapsto z'$\nare $\\mathcal{A}$-terms. Finally, if $S\\neq\\emptyset$ then $\\nu$\nis a blow-up chart and $\\nu\\left(S\\times\\mathbb{R}\\right)$ is the\ncommon zeroset of two variables.\\end{lem}\n\\begin{proof}\nWe only give the details for $\\nu:\\left(x',y',z'\\right)\\mapsto\\left(x',y',x_{1}'\\left(\\lambda+z'\\right)\\right)$,\nfor some $\\lambda\\in\\mathbb{R}$. In this case, $\\nu_{0}$ is the\nidentity map, $S=\\left\\{ x_{1}'=0\\right\\} $ and $\\nu\\left(S\\times\\mathbb{R}\\right)=\\left\\{ x_{1}=z=0\\right\\} $.\nFor all $\\left(x',y'\\right)\\not\\in S$, the inverse function $z\\mapsto z'=\\frac{z}{x_{1}'}-\\lambda$\nis an $\\mathcal{A}$-term. $ $\\end{proof}\n\\begin{defn}\nWe say that an admissible family $\\mathcal{F}$ of transformations\n$\\left(x',y',z'\\right)\\mapsto\\left(x,y,z\\right)$ \\emph{respects }$z$\nif all the elementary transformations appearing in $\\mathcal{F}$\nrespect $z$ (with the obvious convention that if, for example, $\\mathcal{F}\\ni\\rho=\\nu_{1}\\circ\\nu_{2}:\\left(x',y',z'\\right)\\mapsto\\left(x'',y'',z''\\right)\\mapsto\\left(x,y,z\\right)$,\nthen $\\nu_{1}$ respects $z$ and $\\nu_{2}$ respects $z''$). We\nsay that $\\mathcal{F}$ \\emph{almost} \\emph{respects} $z$ if for\nall $\\rho=\\nu_{1}\\circ\\ldots\\circ\\nu_{k}$ the elementary transformations\n$\\nu_{1},\\ldots,\\nu_{k-1}$ respect $z$ and either $\\nu_{k}$ respects\n$z$ or $\\nu_{k}$ is a blow-up chart at infinity involving $z$ and\nsome other variable (i.e. $\\nu_{k}$ is either $\\pi_{m+1,j}^{\\infty}$\nor $\\pi_{m+n+1,j}^{\\pm\\infty}$, for some $j\\in\\left\\{ 1,\\ldots,m\\right\\} $).\n\\end{defn}\nWe prove the following monomialisation result.\n\\begin{thm}\n\\label{thm: vertical monomialisation}Let $F\\left(X,Y,Z\\right)\\in\\widehat{\\mathcal{C}}_{m,n,1}$.\nThen, after admissible family almost respecting $Z$, we have that\n$F$ is normal. \n\\end{thm}\nBefore proving the above theorem, we show how it implies the Main\nTheorem. Since we want to keep track of standard and generalised variables,\nwe will change the notation and prove the Main Theorem for a germ\n$f\\left(x,y,z\\right)\\in\\mathcal{C}_{m,n,1}$, where $y$ is now an\n$n$-tuple of variables and $z$ is a single variable.\n\\begin{proof}\n[Proof of the Main Theorem]Let $f\\left(x,y,z\\right)\\in\\mathcal{C}_{m,n,1}$.\nBy Theorem \\ref{thm: vertical monomialisation} and the quasianalyticity\nproperty, after some admissible family almost respecting $z$, the\ngerm of $f$ is normal (i.e. it is the product of a monomial by a\nunit of $\\mathcal{C}$). The proof is by induction on the pairs $\\left(d,l\\right)$,\nwhere $d=m+n+1$ is the total number of variables and $l$ is the\nminimal length of an admissible monomialising family for $f$. \n\nIf $d=0$ or $l=0$ then there is nothing to prove. So we may suppose\n$d,l>0$.\n\nLet $\\mathcal{F}$ be a monomialising family for $f$ of length $l$.\nNote that, for every $\\rho\\in\\mathcal{F}$, we may partition the domain\nof $\\rho$ (which is either of the form $I_{m_{\\rho}+1,n_{\\rho},r_{\\rho}}$\nor $I_{m_{\\rho},n_{\\rho}+1,r_{\\rho}}$, for some $m_{\\rho},n_{\\rho}$\nsuch that $m_{\\rho}+n_{\\rho}=m+n$) into a finite union of sub-quadrants\n$Q_{\\rho,j}$ (i.e. sets of the form $B_{1}\\times\\ldots\\times B_{m+n+1}$,\nwhere $B_{i}$ is either $\\left\\{ 0\\right\\} $, or $\\left(-r_{\\rho,i},0\\right)$,\nor $\\left(0,r_{\\rho,i}\\right)$) such that $f\\circ\\rho$ has constant\nsign on $Q_{\\rho,j}$. By a classical compactness argument (see for\nexample \\cite[p. 4406]{vdd:speiss:gen}), there exists a finite subfamily\n$\\mathcal{F}_{0}\\subseteq\\mathcal{F}$ and an open neighbourhood $W\\subseteq\\mathbb{R}^{m+n+1}$\nof the origin such that $W\\cap\\text{dom}\\left(f\\right)=\\bigcup_{\\rho\\in\\mathcal{F}_{0}}\\bigcup_{j\\leq J}\\rho\\left(Q_{\\rho,j}\\right)$\n, for some $J\\in\\mathbb{N}$. Notice that, if $A,B$ are $\\mathcal{A}$-cells,\nthen $A\\cap B$ and $A\\setminus B$ are finite disjoint unions of\n$\\mathcal{A}$-cells. \n\nLet $\\mathcal{F}_{1}$ be an elementary family and $\\mathcal{F}_{2}$\nbe an admissible family of length $\\ldots>\\alpha_{d}\\in\\mathbb{K}$,\n$H_{1},\\ldots,H_{d}\\in\\widehat{\\mathcal{C}}_{m,n}$, which are normal,\nand units $U_{1},\\ldots,U_{d}\\in\\left(\\widehat{\\mathcal{C}}_{m,n,1}\\right)^{\\times}$\nsuch that $F\\left(X,Y,Z\\right)=H_{1}\\left(X,Y\\right)G\\left(X,Y,Z\\right)$,\nwhere \n\\[\nG\\left(X,Y,Z\\right)=Z^{\\alpha_{1}}U_{1}\\left(X,Y,Z\\right)+H_{2}\\left(X,Y\\right)Z^{\\alpha_{2}}U_{2}\\left(X,Y,Z\\right)+\\ldots+H_{d}\\left(X,Y\\right)Z^{\\alpha_{d}}U_{d}\\left(X,Y,Z\\right).\n\\]\n\\end{defn}\n\\begin{prop}\n\\label{prop: finite pres}Suppose that the Inductive Hypothesis \\ref{empty: ind hyp}\nholds. Then $F$ admits a finite presentation of some order $d\\in\\mathbb{N}$,\nafter admissible family respecting the variable $Z$ (in fact, the\nadmissible transformations required act as the identity on $Z$).\n\\end{prop}\nThe ring $\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $ is\nclearly not Noetherian. However, the next lemma provides a finiteness\nproperty which is enough for our purposes. The proof takes inspiration\nfrom \\cite[Theorem 6.3.3]{horm}. \n\\begin{lem}\n\\label{lem: quasi noeth}Let $\\mathcal{G}=\\left\\{ F_{\\alpha}\\left(X,Y\\right):\\ \\alpha\\in A\\right\\} \\subseteq\\widehat{\\mathcal{C}}_{m,n}$\nbe a family with good total support. Then,\\smallskip{}\n\n\n\\noindent \\begin{flushleft}\na) after admissible family, there are $\\beta\\in[0,\\infty)^{m}$ and\na collection $\\left\\{ G_{\\alpha}\\left(X,Y\\right):\\ \\alpha\\in A\\right\\} \\subseteq\\widehat{\\mathcal{C}}_{m,n}$\nsuch that $\\forall\\alpha\\in A,\\ F_{\\alpha}\\left(X,Y\\right)=X^{\\beta}G_{\\alpha}\\left(X,Y\\right)$\nand $G_{\\alpha_{0}}\\left(0,Y\\right)\\not\\equiv0$, for some $\\alpha_{0}\\in A$;\n\\par\\end{flushleft}\n\n\\noindent \\begin{flushleft}\nb) for every $d\\in\\mathbb{N}$, after admissible family, the $\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $-module\ngenerated by the tuples $\\left\\{ \\left(F_{\\alpha_{1}},\\ldots,F_{\\alpha_{d}}\\right):\\ \\alpha_{1},\\ldots,\\alpha_{d}\\in A\\right\\} $\nis finitely generated.\n\\par\\end{flushleft}\n\nThe numbers $m,n$ may change after admissible transformation.\\end{lem}\n\\begin{proof}\nFor the proof of a), we view $\\mathcal{G}$ as a subset of $\\mathbb{B}\\left\\llbracket X^{*}\\right\\rrbracket $,\nwith $\\mathbb{B}=\\mathbb{R}\\left\\llbracket Y\\right\\rrbracket $. In\n\\cite[4.11]{vdd:speiss:gen} the authors define the \\emph{blow-up\nheight} of a finite set of monomials, denoted by $b_{X}$. It follows\nfrom the definition of $b_{X}$ that if $b_{X}\\left(\\mathcal{G}_{\\mathrm{min}}\\right)=\\left(0,0\\right)$,\nthen there exists $\\beta\\in[0,\\infty)^{m}$ such that $\\mathcal{G}_{\\mathrm{min}}=\\left\\{ X^{\\beta}\\right\\} $,\nwhich is what we want. The proof of this step is by induction on the\npairs $\\left(m,b_{X}\\left(\\mathcal{G}_{\\text{min}}\\right)\\right)$,\nordered lexicographically. If $m=0$, there is nothing to prove. If\n$m=1$, then $b_{X}\\left(\\mathcal{G}_{\\mathrm{min}}\\right)=\\left(0,0\\right)$. \n\nHence we may assume that $m>1$ and $b_{X}\\left(\\mathcal{G}_{\\mathrm{min}}\\right)\\not=\\left(0,0\\right)$.\nIt follows from the proof of \\cite[Proposition 4.14]{vdd:speiss:gen}\nthat there are $i,j\\in\\left\\{ 1,\\ldots,m\\right\\} $ and suitable ramifications\n$r_{i}^{\\gamma},\\ r_{j}^{\\delta}$ of the variables $X_{i}$ and $X_{j}$\nsuch that, after the admissible transformations $\\rho_{0}:=r_{i}^{\\gamma}\\circ r_{j}^{\\delta}\\circ\\mathfrak{\\pi}_{i,j}^{0}$\nand $\\rho_{\\infty}:=r_{i}^{\\gamma}\\circ r_{j}^{\\delta}\\circ\\pi_{i,j}^{\\infty}$,\nthe blow-up height $b_{X}$ of $\\mathcal{G}_{\\text{min}}$ has decreased\n(to see this, consider $\\alpha_{i},\\beta_{j}$ in the proof of \\cite[Lemma 4.10]{vdd:speiss:gen}\nand perform the mentioned ramifications with $\\gamma=\\beta_{j}$ and\n$\\delta=\\alpha_{i}$). Moreover, for every $\\lambda\\in\\left(0,\\infty\\right)$,\nafter the admissible transformation $\\rho_{\\lambda}:=r_{i}^{\\gamma}\\circ r_{j}^{\\delta}\\circ\\pi_{i,j}^{\\lambda}$,\nthe series in the family $\\mathcal{G}$ have one less generalised\nvariable and one more standard variable, so $m$ has decreased. Since\nadmissible transformations preserve having good total support, the\ninductive hypothesis applies and we obtain the required conclusion.\\bigskip{}\n\n\n\nThe proof of b) is by induction on the pairs $\\left(m+n,d\\right)$,\nordered lexicographically. Arguing by induction on $d$ as in \\cite[Lemma 6.3.2]{horm},\nit is enough to prove the case $d=1$. If $m+n=1$ then, since $\\mathcal{G}$\nhas good total support, the ideal generated by $\\mathcal{G}$ is principal.\nHence suppose that $m+n>1$. Recall that, by part a) of this lemma,\nthere are $\\beta\\in[0,\\infty)^{m}$ and a collection $\\left\\{ G_{\\alpha}\\left(X,Y\\right):\\ \\alpha\\in A\\right\\} \\subseteq\\widehat{\\mathcal{C}}_{m,n}$\nsuch that $\\forall\\alpha\\in A,\\ F_{\\alpha}\\left(X,Y\\right)=X^{\\beta}G_{\\alpha}\\left(X,Y\\right)$\nand $G_{\\alpha_{0}}\\left(0,Y\\right)\\not\\equiv0$, for some $\\alpha_{0}\\in A$.\nAfter a linear transformation $L_{n,c}$, we may suppose that $G_{\\alpha_{0}}$\nis regular of some order $d$ in the variable $Y_{n}$.\n\n\nLet $\\hat{Y}=\\left(Y_{1},\\ldots,Y_{n-1}\\right)$. By the formal Weierstrass\nDivision for generalised power series (see \\cite[4.17]{vdd:speiss:gen}),\nfor every $\\alpha\\in A$ there are $Q_{\\alpha}\\in\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nand $B_{\\alpha,0},\\ldots,B_{\\alpha,d-1}\\in\\mathbb{R}\\left\\llbracket X^{*},\\hat{Y}\\right\\rrbracket $\nsuch that $G_{\\alpha}=G_{\\alpha_{0}}Q_{\\alpha}+R_{\\alpha}$, where\n$R_{\\alpha}\\left(X,Y\\right)=\\sum_{i=0}^{d-1}B_{\\alpha,i}\\left(X,\\hat{Y}\\right)Y_{n}^{i}$.\nIt is proven in \\cite[p. 4390]{vdd:speiss:gen} that the total support\nof the collection $\\left\\{ B_{\\alpha,j}:\\ \\alpha\\in A,\\ j=0,\\ldots,d-1\\right\\} $\nis contained in the good set $\\Sigma\\text{Supp}\\left(\\mathcal{G}\\right)$\nof all finite sums (done component-wise) of elements of $\\text{Supp}\\left(\\mathcal{G}\\right)$.\nHence, by the inductive hypothesis on the total number of variables,\nafter admissible family acting on $\\left(X,\\widehat{Y}\\right)$, the\n$\\mathbb{R}\\left\\llbracket X^{*},\\hat{Y}\\right\\rrbracket $-module\ngenerated by $\\mathcal{B}=\\left\\{ B_{\\alpha}=\\left(B_{\\alpha,0},\\ldots,B_{\\alpha,d-1}\\right):\\ \\alpha\\in A\\right\\} $\nis finitely generated. Therefore, there are $\\alpha_{1},\\ldots,\\alpha_{q}\\in A$\nand for all $\\alpha\\in A$ there are $C_{\\alpha,1},\\ldots,C_{\\alpha,q}\\in\\mathbb{R}\\left\\llbracket X^{*},\\hat{Y}\\right\\rrbracket $\nsuch that $B_{\\alpha}=\\sum_{j=1}^{q}C_{\\alpha,j}B_{\\alpha_{j}}$.\nPutting everything together, we obtain that, for every $\\alpha\\in A$,\n\\[\nF_{\\alpha}=\\left(Q_{\\alpha}-\\sum_{j=1}^{q}C_{\\alpha,j}Q_{\\alpha_{j}}\\right)F_{\\alpha_{0}}+\\sum_{j=1}^{q}C_{\\alpha,j}F_{\\alpha_{j}}.\n\\]\n\n\n\\end{proof}\n\n\\begin{proof}\n[Proof of Proposition \\ref{prop: finite pres}]Write $F\\left(X,Y,Z\\right)=\\sum_{\\alpha\\in A}F_{\\alpha}\\left(X,Y\\right)Z^{\\alpha}$\nand consider the family $\\mathcal{G}=\\left\\{ F_{\\alpha}\\left(X,Y\\right):\\ \\alpha\\in A\\right\\} $,\nwhich is contained in $\\widehat{\\mathcal{C}}_{m,n}$ by Conditions\n2 and 5 in \\ref{emp:properties of the morph}. Note that $A\\subseteq[0,\\infty)$\nis a well ordered set and $\\mathcal{G}$ has good total support.\n\nBy Lemma \\ref{lem: quasi noeth}, after admissible family acting on\n$\\left(X,Y\\right)$, the $\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $-ideal\ngenerated by $\\mathcal{G}$ is finitely generated. Hence we can apply\nthe Inductive Hypothesis \\ref{empty: ind hyp} simultaneously to the\ngenerators and obtain that, after admissible family acting on $\\left(X,Y\\right)$,\nthe generators are normal and linearly ordered by division. Hence,\nthere is $\\alpha_{1}\\in A$ and for all $\\alpha\\in A$ there is $Q_{\\alpha}\\in\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nsuch that $F_{\\alpha}=F_{\\alpha_{1}}\\cdot Q_{\\alpha}$. Notice that,\nsince $F_{\\alpha_{1}}$ is normal, by monomial division $Q_{\\alpha}\\in\\widehat{\\mathcal{C}}_{m,n}$\n(by Remark \\ref{rem: taylor}, the inverse of a unit belonging to\n$\\hat{\\mathcal{C}}$ also belongs to $\\hat{\\mathcal{C}}$). This allows\nus to write \n\\[\nF\\left(X,Y,Z\\right)=\\sum_{\\alpha<\\alpha_{1}}F_{\\alpha}\\left(X,Y\\right)Z^{\\alpha}+F_{\\alpha_{1}}\\left(X,Y\\right)Z^{\\alpha_{1}}U\\left(X,Y,Z\\right),\n\\]\nwhere $U\\left(X,Y,Z\\right)=1+\\sum_{\\alpha>\\alpha_{1}}Q_{\\alpha}\\left(X,Y\\right)Z^{\\alpha-\\alpha_{1}}$.\nThe series $G\\left(X,Y,Z\\right)=\\sum_{\\alpha<\\alpha_{1}}F_{\\alpha}\\left(X,Y\\right)Z^{\\alpha}$\nbelongs to $\\widehat{\\mathcal{C}}_{m,n,1}$ by Condition 5 in \\ref{emp:properties of the morph},\nhence $U\\in\\left(\\widehat{\\mathcal{C}}_{m,n,1}\\right)^{\\times}$.\nWe repeat the above argument for $G$. This procedure will provide,\nafter admissible family acting on $\\left(X,Y\\right)$, a decreasing\nsequence $\\alpha_{1}>\\alpha_{2}>\\ldots$ which is necessarily finite\n(say, of length $d$), since $A$ is well-ordered. Now it is enough\nto rename $H_{i}:=Q_{\\alpha_{i}}$ for $i=1,\\ldots,d$ and factor\nout $H_{1}$ to obtain the required finite presentation.\n\n\\end{proof}\nWe can now finish the proof of Theorem \\ref{thm: vertical monomialisation}\nby showing how to reduce the order of a finite presentation for $F$.\n\\begin{proof}\n[Proof of theorem \\ref{thm: vertical monomialisation}]In what follows,\nup to suitable reflections, there is no harm in considering the variables\n$\\left(X,Y\\right)$ as generalised, hence, to simplify the notation,\nwe will suppose $Y=\\emptyset$.\n\n\\bigskip{}\n\n\nSuppose first that $F\\in\\widehat{\\mathcal{C}}_{m,1}$, i.e. $Z$ is\na standard variable. By Proposition \\ref{prop: finite pres}, we may\nsuppose that $F$ admits a finite presentation as in Definition \\ref{def: finite pres}.\nSince the exponents $\\alpha_{i}$ are in $\\mathbb{N}$, we have that\n$G$ is regular of order $\\alpha_{1}$ in the variable $Z$.\n\n\nIf $\\alpha_{1}=1$, then we perform the Tschirnhausen transformation\ntranslating $Z$ by the solution to the implicit function problem\n$G=0$, and obtain that $F$ is normal. \n\n\nSuppose that $\\alpha_{1}>1$. We follow, up to suitable reflections\nand ramifications, the algorithm for decreasing the order of regularity\nin the proof of \\cite[Theorem 2.5]{rsw}, which we briefly summarise\n(the details can be found in \\cite[Section 4.2.2]{martin_sanz_rolin_local_monomialization}).\nBy the Taylor formula, there are series $A_{1},\\ldots,A_{d}\\in\\widehat{\\mathcal{C}}_{m}$,\nwith $A_{i}\\left(0\\right)=0$, and a unit $U\\in\\left(\\widehat{\\mathcal{C}}_{m,1}\\right)^{\\times}$\nsuch that\n\\[\nG\\left(X,Z\\right)=A_{d}\\left(X\\right)+\\ldots+A_{1}\\left(X\\right)Z^{\\alpha_{1}-1}+U\\left(X,Z\\right)Z^{\\alpha_{1}}.\n\\]\nAfter a Tschirnhausen translation, we may assume that $A_{1}=0$.\nWe apply the Inductive Hypothesis \\ref{empty: ind hyp} simultaneously\nto the $A_{i}$ in such a way that, after admissible family acting\non $X$, the $A_{i}$ are normal, i.e. $A_{i}\\left(X\\right)=X^{\\beta_{i}}U_{i}\\left(X\\right)$\nfor some $\\beta_{i}\\in\\mathbb{K}^{m}$, $U_{i}\\in\\left(\\widehat{\\mathcal{C}}_{m}\\right)^{\\times}$,\nand for some $l\\in\\left\\{ 2,\\ldots,d\\right\\} $ the series $A_{l}^{1\/l}$\ndivides all the series $A_{i}^{1\/i}$. Let $j\\in\\left\\{ 1,\\ldots m\\right\\} $\nbe such that the variable $X_{j}$ appears with a nonzero exponent\nin the monomial $X^{\\beta_{l}}$ and consider the family of blow-up\ntransformations $\\left\\{ \\pi_{m+1,j}^{\\lambda}:\\ \\lambda\\in\\mathbb{R}\\cup\\left\\{ \\pm\\infty\\right\\} \\right\\} $. \n\n\nAfter the transformations $\\pi_{m+1,j}^{\\pm\\infty}$, the series $G$\nhas the form $Z^{\\alpha_{1}}V\\left(X,Z\\right)$, where $V\\in\\left(\\widehat{\\mathcal{C}}_{m,1}\\right)^{\\times}$,\nso in this case $F$ is normal, and we are done. \n\n\nAfter the transformation $\\pi_{m+1,j}^{0}$, the exponent of $X_{j}$\nin the monomial $X^{\\beta_{l}}$ has decreased by the quantity $l$.\nBy repeating the procedure and applying it to the other variables\nappearing with a nonzero exponent in the monomial $X^{\\beta_{l}}$\n, we can reduce the order of regularity of $G$ to $\\alpha_{1}-l$.\n\n\nFor $\\lambda\\in\\mathbb{R}\\setminus\\left\\{ 0\\right\\} $, after the\ntransformation $\\pi_{m+1,j}^{\\lambda}$, thanks to the fact that $A_{1}=0$,\nthe order of $G$ is at most $\\alpha_{1}-1$. \n\n\nThis shows that, in the case when $Z$ is a standard variable, after\nadmissible family almost respecting $Z$, the series $F$ is normal.\n\\bigskip{}\n\n\n\nNow suppose that $F\\in\\widehat{\\mathcal{C}}_{m+1,0}$, i.e. $Z$ is\na generalised variable. By Proposition \\ref{prop: finite pres}, we\nmay suppose that $F$ admits a finite presentation as in Definition\n\\ref{def: finite pres}. We can apply the Inductive Hypothesis \\ref{empty: ind hyp}\nsimultaneously to $H_{1},\\ldots,H_{d}$ in such a way that, after\nadmissible family, we have\n\\[\nG\\left(X,Z\\right)=Z^{\\alpha_{1}}\\tilde{U}_{1}\\left(X,Z\\right)+X^{\\Gamma_{2}}Z^{\\alpha_{2}}\\tilde{U}_{2}\\left(X,Z\\right)+\\ldots+X^{\\Gamma_{d}}Z^{\\alpha_{d}}\\tilde{U}_{d}\\left(X,Z\\right),\n\\]\nfor some units $\\tilde{U}_{i}\\in\\left(\\widehat{\\mathcal{C}}_{m+1,0}\\right)^{\\times}$,\nand the exponents $\\Gamma_{i}=\\left(\\gamma_{i}^{\\left(1\\right)},\\ldots,\\gamma_{i}^{\\left(m\\right)}\\right)$\nare such that the monomials $\\left\\{ X^{\\frac{\\Gamma_{i}}{\\alpha_{1}-\\alpha_{i}}}:\\ i=2,\\ldots,d\\right\\} $\nare linearly ordered by division. Let $i_{0}\\in\\left\\{ 2,\\ldots,d\\right\\} $\nbe smallest with the property that\n\\[\n\\forall i\\in\\left\\{ 2,\\ldots,d\\right\\} ,\\ \\forall j\\in\\left\\{ 1,\\ldots,m\\right\\} ,\\ \\ \\frac{\\gamma_{i_{0}}^{\\left(j\\right)}}{\\alpha_{1}-\\alpha_{i_{0}}}\\leq\\frac{\\gamma_{i}^{\\left(j\\right)}}{\\alpha_{1}-\\alpha_{i}}.\\tag{\\#}\n\\]\nSuppose $\\gamma_{i_{0}}^{\\left(1\\right)}\\not=0$ and perform a ramification\nof the variable $X_{1}$ with exponent $\\gamma:=\\frac{\\gamma_{i_{0}}^{\\left(1\\right)}}{\\alpha_{1}-\\alpha_{i_{0}}}$.\nWe consider the family of blow-up transformations $\\left\\{ \\pi_{m+1,1}^{\\lambda}:\\ \\lambda\\in\\left[0,\\infty\\right]\\right\\} $. \n\n\nAfter the transformation $\\pi_{m+1,1}^{\\infty}$, we can write\n\\[\nG\\left(X,Z\\right)=Z^{\\alpha_{1}}\\left[\\tilde{U}_{1}\\left(X,Z\\right)+X^{\\Gamma_{2}}Z^{\\beta_{2}}\\tilde{U}_{2}\\left(X,Z\\right)+\\ldots+X^{\\Gamma_{d}}Z^{\\beta_{d}}\\tilde{U}_{d}\\left(X,Z\\right)\\right],\n\\]\nwhere $\\beta_{i}:=\\frac{\\gamma_{i}^{\\left(1\\right)}}{\\gamma_{i_{0}}^{\\left(1\\right)}}\\left(\\alpha_{1}-\\alpha_{i_{0}}\\right)+\\alpha_{i}-\\alpha_{1}$\nis nonnegative, thanks to (\\#). Notice that, since by (\\#) every $\\gamma_{i}^{\\left(1\\right)}$\nis nonzero, the expression between square brackets is a unit. Hence\nin this case $F$ has a finite presentation of order $1$, i.e. $F$\nis normal, and we are done. \n\n\nAfter the transformation $\\pi_{m+1,1}^{0}$, we can write\n\\[\nG\\left(X,Z\\right)=X_{1}^{\\gamma\\alpha_{1}}\\left[Z^{\\alpha_{1}}\\tilde{U}_{1}\\left(X,Z\\right)+X^{\\Delta_{2}}Z^{\\alpha_{2}}\\tilde{U}_{2}\\left(X,Z\\right)+\\ldots+X^{\\Delta_{d}}Z^{\\alpha_{d}}\\tilde{U}_{d}\\left(X,Z\\right)\\right],\n\\]\nwhere $\\Delta_{i}=\\left(\\delta_{i}^{\\left(1\\right)},\\ldots,\\delta_{i}^{\\left(m\\right)}\\right):=\\left(\\gamma_{i}^{\\left(1\\right)}-\\gamma_{i_{0}}^{\\left(1\\right)}\\frac{\\alpha_{1}-\\alpha_{i}}{\\alpha_{1}-\\alpha_{i_{0}}},\\gamma_{i}^{\\left(2\\right)},\\ldots,\\gamma_{i}^{\\left(m\\right)}\\right)$\n. Remark that, by (\\#), the exponents $\\delta_{i}^{\\left(1\\right)}$\nare nonnegative and $\\delta_{i_{0}}^{\\left(1\\right)}=0$. Hence, up\nto factoring out by a power of $X_{1}$, the variable $X_{1}$ does\nnot appear any more in the $i_{0}^{\\text{th}}$ term of the above\nfinite presentation. By repeating this step with the other variables\n$X_{j}$ such that $\\gamma_{i_{0}}^{\\left(j\\right)}\\not=0$, we obtain\n\\[\nG\\left(X,Z\\right)=X^{\\Delta}\\left[Z^{\\alpha_{i_{0}}}V\\left(X,Z\\right)+X^{\\Delta'_{i_{0}+1}}Z^{\\alpha_{i_{0}+1}}\\tilde{U}_{i_{0}+1}\\left(X,Z\\right)+\\ldots+X^{\\Delta_{d}'}Z^{\\alpha_{d}}\\tilde{U}_{d}\\left(X,Z\\right)\\right],\n\\]\nwhere $V\\in\\left(\\widehat{\\mathcal{C}}_{m+1,0}\\right)^{\\times}$,\nthe components of $\\Delta$ are $\\frac{\\alpha_{1}\\gamma_{i_{0}}^{\\left(j\\right)}}{\\alpha_{1}-\\alpha_{i_{0}}}$\nand the components of $\\Delta_{i}'$ are $\\gamma_{i}^{\\left(j\\right)}-\\gamma_{i_{0}}^{\\left(j\\right)}\\frac{\\alpha_{1}-\\alpha_{i}}{\\alpha_{1}-\\alpha_{i_{0}}}$.\nHence $F$ has a finite presentation of order $d-i_{0}+1$.\n\n\nIf $\\lambda\\in\\left(0,\\infty\\right)$, then after the transformation\n$\\pi_{m+1,1}^{\\lambda}$, the variable $Z$ is standard and we have\nreduced to the case $F\\in\\widehat{\\mathcal{C}}_{m,1}$. \n\n\nFinally, notice that if $F\\in\\widehat{\\mathcal{C}}_{0,m+1}$, i.e.\nall the variables are standard, then we can start the proof by first\nramifying the variables $X$ with exponent $d!$, in order to ensure\nthat only natural exponents appear in the series $A_{l}^{1\/l}$.\n\n\\end{proof}\n\\begin{rem}\n\\label{rem: no flatness}In the case when the set of admissible exponents\nis $\\mathbb{N}$ the proof of Theorem \\ref{thm: vertical monomialisation}\ncan be simplified. In fact, by Noetherianity of $\\mathbb{R}\\left\\llbracket X,Y\\right\\rrbracket $,\nthe $\\mathbb{R}\\left\\llbracket X,Y\\right\\rrbracket $-ideal generated\nby the family $\\mathcal{G}$ is finitely generated and one obtains\nimmediately a \\textquotedblleft{}formal\\textquotedblright{} finite\npresentation for $F$, where the units are formal power series, not\nnecessarily belonging to $\\widehat{\\mathcal{C}}$. After monomialising\nthe generators and factoring out an appropriate monomial, this automatically\nimplies that $F$ is regular of some order in the variable $Z$. Hence\nwe can dispense with Proposition \\ref{prop: finite pres} and implement\ndirectly the last part of the proof of Theorem \\ref{thm: vertical monomialisation}. \n\nThis argument also implies that in the real analytic setting, in order\nto obtain regularity in a chosen variable $Z$, there is no need to\nprove a convergent version of the finite presentation in Definition\n\\ref{def: finite pres}. In their proof of quantifier elimination\nfor the real field with restricted analytic functions and the function\n$x\\mapsto1\/x$, Denef and van den Dries prove such a convergent version\n(see \\cite[Lemma 4.12]{vdd:d}), by invoking a consequence of faithful\nflatness in \\cite[(4C)(ii)]{matsumura:commutative_algebra}. Our remark\nimplies that this is not necessary.\n\\end{rem}\n\\bibliographystyle{amsalpha}\n\\def$'${$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{#1}\n\\setcounter{equation}{0}}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{prop}[theorem]{Proposition}\n\n\\theoremstyle{definition}\n\\newtheorem{remark}[theorem]{Remark}\n\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\n\\theoremstyle{definition}\n\\newtheorem{assumption}[theorem]{Assumption}\n\n\\newcommand{{\\int\\hspace*{-4.3mm}\\diagup}}{{\\int\\hspace*{-4.3mm}\\diagup}}\n\\makeatletter\n\\def\\dashint{\\operatorname%\n{\\,\\,\\text{\\bf-}\\kern-.98em\\DOTSI\\intop\\ilimits@\\!\\!}}\n\\makeatother\n\n\\newcommand{\\WO}[2]{\\overset{\\scriptscriptstyle0}{W}\\,\\!^{#1}_{#2}}\n\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\\def\\textit{\\textbf{c}}{\\textit{\\textbf{c}}}\n\\def\\textit{\\textbf{u}}{\\textit{\\textbf{u}}}\n\\def\\textit{\\textbf{v}}{\\textit{\\textbf{v}}}\n\\def\\textit{\\txfextbf{w}}{\\textit{\\txfextbf{w}}}\n\\def\\textit{\\textbf{f}}{\\textit{\\textbf{f}}}\n\\def\\textit{\\textbf{g}}{\\textit{\\textbf{g}}}\n\\def\\textit{\\textbf{h}}{\\textit{\\textbf{h}}}\n\\def\\textit{\\textbf{P}}{\\textit{\\textbf{P}}}\n\\def\\textit{\\textbf{\\phi}}{\\textit{\\textbf{\\phi}}}\n\\def\\\\det{\\text{det}}\n\n\\def\\tilde{\\mathcal{L}_0^\\sigma}{\\tilde{\\mathcal{L}_0^\\sigma}}\n\\def\\hat{\\mathcal{L}_0^\\sigma}{\\hat{\\mathcal{L}_0^\\sigma}}\n\n\\def\\alpha'+\\sigma{\\alpha'+\\sigma}\n\\def\\alpha'\/\\sigma{\\alpha'\/\\sigma}\n\n\n\\defa{a}\n\\defb{b}\n\\defc{c}\n\n\\def{\\sf A}{{\\sf A}}\n\\def{\\sf B}{{\\sf B}}\n\\def{\\sf M}{{\\sf M}}\n\\def{\\sf S}{{\\sf S}}\n\\def\\mathrm{i}{\\mathrm{i}}\n\n\\def\\.5{\\frac{1}{2}}\n\\def\\mathbb{A}{\\mathbb{A}}\n\\def\\mathbb{O}{\\mathbb{O}}\n\\def\\mathbb{R}{\\mathbb{R}}\n\\def\\mathbb{Z}{\\mathbb{Z}}\n\\def\\mathbb{E}{\\mathbb{E}}\n\\def\\mathbb{N}{\\mathbb{N}}\n\\def\\mathbb{H}{\\mathbb{H}}\n\\def\\mathbb{Q}{\\mathbb{Q}}\n\\def\\mathbb{C}{\\mathbb{C}}\n\n\\def\\tilde{G}{\\tilde{G}}\n\n\\def\\textsl{\\textbf{a}}{\\textsl{\\textbf{a}}}\n\\def\\textsl{\\textbf{x}}{\\textsl{\\textbf{x}}}\n\\def\\textsl{\\textbf{y}}{\\textsl{\\textbf{y}}}\n\\def\\textsl{\\textbf{z}}{\\textsl{\\textbf{z}}}\n\\def\\textsl{\\textbf{w}}{\\textsl{\\textbf{w}}}\n\n\\def\\mathfrak{L}{\\mathfrak{L}}\n\\def\\mathfrak{B}{\\mathfrak{B}}\n\\def\\mathfrak{O}{\\mathfrak{O}}\n\\def\\mathfrak{R}{\\mathfrak{R}}\n\\def\\mathfrak{S}{\\mathfrak{S}}\n\\def\\mathfrak{T}{\\mathfrak{T}}\n\\def\\mathfrak{q}{\\mathfrak{q}}\n\n\\def\\text{Re}\\,{\\text{Re}\\,}\n\\def\\text{Im}\\,{\\text{Im}\\,}\n\n\\def\\mathcal{A}{\\mathcal{A}}\n\\def\\mathcal{B}{\\mathcal{B}}\n\\def\\mathcal{C}{\\mathcal{C}}\n\\def\\mathcal{D}{\\mathcal{D}}\n\\def\\mathcal{E}{\\mathcal{E}}\n\\def\\mathcal{F}{\\mathcal{F}}\n\\def\\mathcal{G}{\\mathcal{G}}\n\\def\\mathcal{H}{\\mathcal{H}}\n\\def\\mathcal{P}{\\mathcal{P}}\n\\def\\mathcal{M}{\\mathcal{M}}\n\\def\\mathcal{O}{\\mathcal{O}}\n\\def\\mathcal{Q}{\\mathcal{Q}}\n\\def\\mathcal{R}{\\mathcal{R}}\n\\def\\mathcal{S}{\\mathcal{S}}\n\\def\\mathcal{T}{\\mathcal{T}}\n\\def\\mathcal{L}{\\mathcal{L}}\n\\def\\mathcal{U}{\\mathcal{U}}\n\\def\\mathcal{I}{\\mathcal{I}}\n\\newcommand\\frC{\\mathfrak{C}}\n\n\\def\\bar{P}{\\bar{P}}\n\n\\newcommand{\\RN}[1]{%\n \\textup{\\uppercase\\expandafter{\\romannumeral#1}}%\n}\n\\newcommand{\\ip}[1]{\\left\\langle#1\\right\\rangle}\n\\newcommand{\\set}[1]{\\left\\{#1\\right\\}}\n\\newcommand{\\norm}[1]{\\lVert#1\\rVert}\n\\newcommand{\\Norm}[1]{\\left\\lVert#1\\right\\rVert}\n\\newcommand{\\abs}[1]{\\left\\lvert#1\\right\\rvert}\n\\newcommand{\\tri}[1]{|\\|#1|\\|}\n\\newcommand{\\operatorname{div}}{\\operatorname{div}}\n\\newcommand{\\text{dist}}{\\text{dist}}\n\\newcommand{\\operatornamewithlimits{argmin}}{\\operatornamewithlimits{argmin}}\n\\renewcommand{\\epsilon}{\\varepsilon}\n\n\\newcounter{marnote}\n\\newcommand\\marginnote[1]{\\stepcounter{marnote}$^{\\bullet\\,\\themarnote}$\\marginpar{\\tiny$\\bullet\\,\\themarnote$:\\,#1}}\n\n\n\n\\begin{document}\n\\title[Asymptotics for the perfect conductivity problem ]{Asymptotics for the electric field when $M$-convex inclusions are close to the matrix boundary}\n\n\n\\author[Z.W. Zhao]{Zhiwen Zhao}\n\n\\address[Z.W. Zhao]{1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. }\n\\address{2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands.}\n\\email{zwzhao@mail.bnu.edu.cn.}\n\n\n\n\n\n\n\n\\date{\\today}\n\n\n\n\n\\begin{abstract}\nIn the perfect conductivity problem of composites, the electric field may become arbitrarily large as $\\varepsilon$, the distance between the inclusions and the matrix boundary, tends to zero. The main contribution of this paper lies in developing a clear and concise procedure to establish a boundary asymptotic formula of the concentration for perfect conductors with arbitrary shape in all dimensions, which explicitly exhibits the singularities of the blow-up factor $Q[\\varphi]$ introduced in \\cite{LX2017} by picking the boundary data $\\varphi$ of $k$-order growth. In particular, the smoothness of inclusions required for at least $C^{3,1}$ in \\cite{LLY2019} is weakened to $C^{2,\\alpha}$, $0<\\alpha<1$ here.\n\\end{abstract}\n\n\n\\maketitle\n\n\n\n\\section{Introduction and main results}\n\nIt is well known that field concentrations appear widely in nature and industrial applications. These fields include extreme electric, heat fluxes and mechanical loads. Motivated by the issue of material failure initiation, in this paper we are devoted to the investigation of blow-up phenomena arising from high-contrast fiber-reinforced composites with the densely packed fibers. The key feature of the concentrated fields is that the blow-up comes from the narrow regions between fibers and the thin gaps between fibers and the matrix boundary. It is worth emphasizing that the latter is more interesting due to the interaction from the boundary data. Although there has made great progress in the engineering and mathematical literature since Babu\\u{s}ka et al's famous work \\cite{BASL1999} over the past two decades, accurate numerical computation of the concentrated field are still very hard for lack of fine characterization to develop an efficient numerical scheme. So, it is significantly important from a practical point of view to precisely describe the singular behavior of such high concentration.\n\nIn the context of electrostatics, the field is the gradient of a solution to the Laplace equation and the blow-up rate of the gradient were captured accurately. Denote the distance between two inclusions or between inclusions and the matrix boundary by $\\varepsilon$. It has been proved that for the perfect conductivity problem, the blow-up rate of the gradient is $\\varepsilon^{-1\/2}$ in two dimensions \\cite{AKLLL2007,BC1984,BLY2009,AKL2005,Y2007,Y2009,K1993}, while it is $|\\varepsilon\\ln\\varepsilon|^{-1}$ in three dimensions \\cite{BLY2009,LY2009,BLY2010,L2012}.\n\nBesides these foregoing estimates of the singularities for the field, there is another direction of investigation to establish the asymptotic formula of $\\nabla u$ in the thin gap of electric field concentration. In two dimensions, consider the following conductivity problem\n\\begin{align}\\label{per001}\n\\begin{cases}\n\\Delta u=0,&\\hbox{in}\\;\\mathbb{R}^{2}\\setminus\\overline{D_{1}\\cup D_{2}},\\\\\nu=C_{j}, &\\hbox{on}\\;\\partial D_{j},\\;j=1,2,\\\\\nu(\\mathbf{x})-H(\\mathbf{x})=O(|\\mathbf{x}|^{-1}),&\\mathrm{as}\\;|\\mathbf{x}|\\rightarrow\\infty,\\\\\n\\int_{\\partial D_{j}}\\frac{\\partial u}{\\partial\\nu}\\big|_{+}=0,&j=1,2,\n\\end{cases}\n\\end{align}\nwhere $H$ is a given harmonic function in $\\mathbb{R}^{2}$ and\n$$\\frac{\\partial u}{\\partial\\nu}\\Big|_{+}:=\\lim_{\\tau\\rightarrow0}\\frac{u(x+\\nu\\tau)-u(x)}{\\tau}.$$\nHere and throughtout this paper $\\nu$ is the unit outer normal of $D_{j}$ and the subscript $\\pm$ shows the limit from outside and inside the domain, respectively. For problem (\\ref{per001}), Kang, Lim and Yun \\cite{KLY2013} obtained a complete characterization of the singularities of $\\nabla u$ with $D_{1}$ and $D_{2}$ being disks as follows\n\\begin{align}\\label{singular}\n\\nabla u(\\mathbf{x})=\\frac{2r_{1}r_{2}}{r_{1}+r_{2}}(\\mathbf{n}\\cdot\\nabla H)(\\mathbf{p})\\nabla h(\\mathbf{x})+\\nabla g(\\mathbf{x}),\n\\end{align}\nwhere $h(\\mathbf{x})=\\frac{1}{2\\pi}(\\ln|\\mathbf{x}-\\mathbf{p}_{1}|-\\ln|\\mathbf{x}-\\mathbf{p}_{2}|)$ with $\\mathbf{p}_{1}\\in D_{1}$ and $\\mathbf{p}_{2}\\in D_{2}$ being the fixed point of $R_{1}R_{2}$ and $R_{2}R_{1}$ respectively, $R_{j}$ is the reflection with respect to $\\partial D_{j}$, $\\mathbf{n}$ is the unit vector in the direction of $\\mathbf{p}_{2}-\\mathbf{p}_{1}$, $\\mathbf{p}$ is the middle point of the shortest line segment connecting $\\partial D_{1}$ and $\\partial D_{2}$, and $|\\nabla g|$ is bounded independently of $\\varepsilon$ on any bounded subset of $\\mathbb{R}^{2}\\setminus\\overline{D_{1}\\cup D_{2}}$. Obviously $\\nabla h$ characterizes the singular behavior of $\\nabla u$ explicitly. Ammari, Ciraolo, Kang, Lee, Yun \\cite{ACKLY2013} extended the characterization (\\ref{singular}) to the case when inclusions $D_{1}$ and $D_{2}$ are strictly convex domains in $\\mathbb{R}^{2}$ by utilizing disks osculating to convex domains. In three dimensions, Kang, Lim and Yun \\cite{KLY2014} derived an asymptotic formula of $\\nabla u$ for two spherical perfect conductors with the same radii. The asymptotics for perfectly conducting particles with the different radii can be seen in \\cite{LWX2019}. Recently, a great work on establishing an asymptotic formula in dimensions two and three for two arbitrarily $2$-convex inclusions has been completed by Li, Li and Yang in \\cite{LLY2019}. It is worth mentioning that for high-contrast composites with the matrix described by nonlinear constitutive laws such as $p$-Laplace, Gorb and Novikov \\cite{G2012} captured the stress concentration factor. Additionally, the asymptotics of the eigenvalues of the Poincar\\'{e} variational problem for two close-to-touching inclusions were obtained by Bonnetier and Triki in \\cite{BT2013}. More related work can be seen in \\cite{ABTV2015,AKLLZ2009,BCN2013,BLL2015,BLL2017,BT2012,BV2000,DL2019,G2015,KY2019,KLY2015,LLBY2014,LY2015,M1996,MMN2007,BJL2017,LX2017}.\n\nHowever, to the best of our knowledge, previous investigations on the asymptotics of the field concentration only focused on the narrow region between inclusions. This paper, by contrast, aims at deriving a completely asymptotic characterization for the perfect conductivity problem (\\ref{con002}) with $m$-convex inclusions close to the matrix boundary and the boundary data of $k$-order growth in all dimensions. The asymptotic results in this paper also provide an efficient way to compute the electrical field numerically.\n\nTo state our main works in a precise manner, we first describe our domain and notations. Let $D\\subset\\mathbb{R}^{n}\\,(n\\geq2)$ be a bounded domain with $C^{2,\\alpha}~(0<\\alpha<1)$ boundary, which has a $C^{2,\\alpha}$-subdomain $D_{1}^{\\ast}$ touching matrix boundary $\\partial D$ only at one point. That is, by a translation and rotation of the coordinates, if necessary,\n\\begin{align*}\n\\partial D_{1}^{\\ast}\\cap\\partial D=\\{0'\\}\\subset\\mathbb{R}^{n-1}.\n\\end{align*}\nThroughout the paper, we use superscript prime to denote ($n-1$)-dimensional domains and variables, such as $\\Sigma'$ and $x'$. After a translation, we set\n\\begin{align*}\nD_{1}^{\\varepsilon}:=D_{1}^{\\ast}+(0',\\varepsilon),\n\\end{align*}\nwhere $\\varepsilon>0$ is a sufficiently small constant. For the sake of simplicity, denote\n\\begin{align*}\nD_{1}:=D_{1}^{\\varepsilon},\\quad\\mathrm{and}\\quad\\Omega:=D\\setminus\\overline{D}_{1}.\n\\end{align*}\n\nThe conductivity problem with inclusions close to touching matrix boundary can be modeled by the following scalar equation with piecewise constant coefficients\n\\begin{align}\\label{con001}\n\\begin{cases}\n\\mathrm{div}{(a_{k}(x)\\nabla u)}=0,&\\hbox{in}\\;D,\\\\\nu=\\varphi, &\\hbox{on}\\;\\partial{D},\n\\end{cases}\n\\end{align}\nwhere\n\\begin{align*}\na_{k}(x)=\n\\begin{cases}\nk\\in[0,1)\\cup(1,\\infty],&\\hbox{in}\\;D_{1},\\\\\n1,&\\hbox{on}\\;D\\setminus D_{1}.\n\\end{cases}\n\\end{align*}\nActually, equation (\\ref{con001}) can also be used to describe more physical phenomenon, such as dielectrics, magnetism, thermal conduction, chemical diffusion and flow in porous media.\n\nWhen the conductivity of $D_{1}$ degenerates to be infinity, problem (\\ref{con001}) turns into the perfect conductivity problem as follows\n\\begin{align}\\label{con002}\n\\begin{cases}\n\\Delta u=0,&\\hbox{in}\\;D\\setminus D_{1},\\\\\nu=C_{1}, &\\hbox{in}\\;\\overline{D}_{1},\\\\\n\\int_{\\partial D_{1}}\\frac{\\partial u}{\\partial\\nu}\\big|_{+}=0,\\\\\nu=\\varphi, &\\mathrm{on}\\;\\partial D,\n\\end{cases}\n\\end{align}\nwhere the free constant $C_{1}$ is determined later by the third line of (\\ref{con002}). There has established the existence, uniqueness and regularity of weak solutions to (\\ref{con002}) in \\cite{BLY2009} with a minor modification. We further assume that there exists a small constant $R>0$ independent of $\\varepsilon$, such that the portions of $\\partial D$ and $\\partial D_{1}$ near the origin can be written as\n\\begin{align*}\nx_{n}=\\varepsilon+h_{1}(x')\\quad\\mathrm{and}\\quad x_{n}=h(x'),\\quad\\quad x'\\in B_{2R}',\n\\end{align*}\nwhere $h_{1}$ and $h$ satisfy that for $m\\geq2$,\n\\begin{enumerate}\n{\\it\\item[(\\bf{\\em H1})]\n$h_{1}(x')-h(x')=\\lambda|x'|^{m}+O(|x'|^{m+1}),$\n\\item[(\\bf{\\em H2})]\n$|\\nabla_{x'}^{i}h_{1}(x')|,\\,|\\nabla_{x'}^{i}h(x')|\\leq \\kappa_{1}|x'|^{m-i},\\;\\,i=1,2,$\n\\item[(\\bf{\\em H3})]\n$\\|h_{1}\\|_{C^{2,\\alpha}(B'_{2R})}+\\|h\\|_{C^{2,\\alpha}(B'_{2R})}\\leq \\kappa_{2},$}\n\\end{enumerate}\nwhere $\\lambda$ and $\\kappa_{j},j=1,2$, are three positive constants independent of $\\varepsilon$.\n\nTo explicitly uncover the effect of boundary data $\\varphi$ on the singularities of the field, we classify $\\varphi\\in C^{2}(\\partial D)$ according to its parity as follows. Denote the bottom boundary of $\\Omega_{R}$ by $\\Gamma^{-}_{R}=\\{x\\in\\mathbb{R}^{n}|\\,x_{n}=h(x'),\\,|x'|0$ and $k>1$ is a positive integer.\n\nFor $z'\\in B'_{R},\\,0n+i-1,\\\\\n|\\ln\\varepsilon|,&m=n+i-1,\\\\\n1,&mn+i-1,\\\\\n1,&m=n+i-1,\n\\end{cases}\n\\end{align*}\nwhere $\\Gamma(s)=\\int^{+\\infty}_{0}t^{s-1}e^{-t}\\,dt$, $s>0$ is the Gamma function. Denote by $\\omega_{n-1}$ the area of the surface of unit sphere in $(n-1)$-dimension. For $(z',z_{n})\\in\\Omega_{2R}$, denote\n\\begin{align}\\label{ZZW666}\n\\delta(z'):=\\varepsilon+h_{1}(z')-h(z').\n\\end{align}\nLet $\\Omega^{\\ast}:=D\\setminus\\overline{D^{\\ast}_{1}}$. We define a linear functional with respect to $\\varphi$,\n\\begin{align}\\label{linear001}\nQ^{\\ast}[\\varphi]:=\\int_{\\partial D_{1}}\\frac{\\partial v_{0}^{\\ast}}{\\partial\\nu},\n\\end{align}\nwhere $v_{0}^{\\ast}$ is a solution of the following problem:\n\\begin{align}\\label{con003}\n\\begin{cases}\n\\Delta v_{0}^{\\ast}=0,\\quad\\quad\\;\\,&\\mathrm{in}\\;\\Omega^{\\ast},\\\\\nv_{0}^{\\ast}=0,\\quad\\quad\\;\\,&\\mathrm{on}\\;\\partial D_{1}^{\\ast},\\\\\nv_{0}^{\\ast}=\\varphi(x),\\quad\\;\\,&\\mathrm{on}\\;\\partial D.\n\\end{cases}\n\\end{align}\nNote that the definition of $Q^{\\ast}[\\varphi]$ is valid under case ({\\bf{\\em S2}}) but only valid for $m1$ are all positive integers in the following.\n\\begin{theorem}\\label{thm001}\nAssume that $D_{1}\\subset D\\subseteq\\mathbb{R}^{n}\\,(n\\geq2)$ are defined as above, conditions ({\\bf{\\em H1}})--({\\bf{\\em H3}}) and ({\\bf{\\em S1}}) hold. Let $u\\in H^{1}(D;\\mathbb{R}^{n})\\cap C^{1}(\\overline{\\Omega};\\mathbb{R}^{n})$ be the solution of (\\ref{con002}). Then for a sufficiently small $\\varepsilon>0$ and $x\\in\\Omega_{R}$,\n\n$(i)$ for $m\\geq n+k-1$,\n\\begin{align*}\n\\nabla u=\\frac{\\eta\\Gamma^{n+k}_{m}}{\\lambda^{\\frac{k}{m}}\\Gamma^{n}_{m}}(1+O(r_{\\varepsilon}))\\rho_{k;0}(n,m;\\varepsilon)\\nabla\\bar{u}+\\nabla\\bar{u}_{0}+O(\\mathbf{1})\\delta^{1-\\frac{2}{m}}\\|\\varphi\\|_{C^{2}(\\partial D)};\n\\end{align*}\n\n$(ii)$ for $n-1\\leq mn+k,\\\\\n\\varepsilon^{\\frac{1}{m}}|\\ln\\varepsilon|,&m=n+k,\\\\\n|\\ln\\varepsilon|^{-1},&m=n+k-1,\\\\\n\\varepsilon^{\\frac{n+k-1-m}{(n+k-1)(m+1)}},&n-1n+k-1$ or $|\\ln\\varepsilon|$ if $m=n+k-1$ for the boundary data $\\varphi$ with $k$-order growth. In addition, when $m>2$, the remainder of order $O(\\varepsilon^{1-2\/m})$ in the shortest line segment between the conductors and the matrix boundary provides a more precise characterization on the asymptotic behavior of the concentration than that of $m=2$. Finally, the concisely main terms $\\nabla\\bar{u}$ and $\\nabla\\bar{u}_{0}$ together with their coefficients can completely describe the singular effect of the geometry, which will greatly reduce the complexity of numerical computation for $\\nabla u$.\n\\end{remark}\n\n\\begin{remark}\nThe asymptotics of $\\nabla u$ in Theorem \\ref{thm001} indicate that\n\n$(1)$ if $m\\leq n+k-1$, then its maximum achieves only at $\\{x'=0'\\}\\cap\\Omega$;\n\n$(2)$ if $m>n+k-1$, then the maximum achieves at both $\\{x'=0'\\}\\cap\\Omega$ and $\\{|x'|=\\varepsilon^{\\frac{1}{m}}\\}\\cap\\Omega$.\n\\end{remark}\n\n\n\\begin{remark}\nIn order to further reveal the effect of principal curvatures of the geometry, we take $n=3$ relevant to physical dimension for example. Consider\n\\begin{align*}\n\\varphi=\\eta_{1}|x_{1}|^{k}+\\eta_{2}|x_{2}|^{k},\\quad x\\in\\{\\lambda_{1}|x_{1}|^{m}+\\lambda_{2}|x_{2}|^{m}0$,\n\n$(i)$ for $m\\geq n-1$,\n\\begin{align*}\n\\nabla u=\\frac{m\\lambda^{\\frac{n-1}{m}}Q^{\\ast}[\\varphi]}{(n-1)\\omega_{n-1}\\Gamma^{n}_{m}}\\frac{1+O(\\tilde{r}_{\\varepsilon})}{\\rho_{0}(n,m;\\varepsilon)}\\nabla\\bar{u}+\\nabla\\bar{u}_{0}+O(\\mathbf{1})\\delta^{1-\\frac{2}{m}}\\|\\varphi\\|_{C^{2}(\\partial D)};\n\\end{align*}\n\n$(ii)$ for $mn-1,\\\\\n|\\ln\\varepsilon|^{-1},&m=n-1,\\\\\n\\max\\{\\varepsilon^{\\frac{m+n-2}{(m+1)(2m+n-2)}},\\varepsilon^{\\frac{1}{6}}\\}.&mn$, then the maximum attains at $\\{|x'|=\\varepsilon^{\\frac{1}{m}}\\}\\cap\\Omega$.\n\\end{remark}\n\n\\begin{remark}\nIf (\\ref{Geometry}) holds in Theorem \\ref{coro002}, we can obtain that the coefficient of the main term $\\nabla\\bar{u}$ has an explicit dependence of $\\sqrt[m]{\\lambda_{1}\\lambda_{2}}$.\n\\end{remark}\n\nThe organization of this paper is as follows. In section 2, we carry out a linear decomposition of the solution $u$ to problem (\\ref{con002}) as $v_{0}$ and $v_{1}$, defined by (\\ref{con005}) and (\\ref{con006}) below, and we prove the correspondingly main terms $\\bar{u}_{0}$ and $\\bar{u}$ constructed by (\\ref{con009}) and (\\ref{con016}), respectively, in Lemma \\ref{lem001} and Theorem \\ref{thm002}. Based on the results obtained in section 2, we give the proofs of Theorem \\ref{thm001} and Theorem \\ref{coro002} consisting of the asymptotics of blow-up factor $Q[\\varphi]$ and $a_{11}$ in section 3.\n\n\n\n\n\n\n\n\n\\section{Preliminary}\n\n\\subsection{Solution split}\nAs in \\cite{LX2017}, we decompose the solution $u$ of (\\ref{con002}) as follows\n\\begin{align}\\label{con0033}\nu(x)=C_{1}v_{1}(x)+v_{0}(x),\\quad\\;\\,\\mathrm{in}\\;D\\setminus\\overline{D}_{1},\n\\end{align}\nwhere $v_{i}$, $i=0,1$, verify\n\\begin{align}\\label{con005}\n\\begin{cases}\n\\Delta v_{0}=0,\\quad\\quad\\;\\,&\\mathrm{in}\\;\\Omega,\\\\\nv_{0}=0,\\quad\\quad\\;\\,&\\mathrm{on}\\;\\partial D_{1},\\\\\nv_{0}=\\varphi(x),\\quad\\;\\,&\\mathrm{on}\\;\\partial D,\n\\end{cases}\n\\end{align}\nand\n\\begin{align}\\label{con006}\n\\begin{cases}\n\\Delta v_{1}=0,\\quad\\quad\\;\\,&\\mathrm{in}\\;\\Omega,\\\\\nv_{1}=1,\\quad\\quad\\;\\,&\\mathrm{on}\\;\\partial D_{1},\\\\\nv_{1}=0,\\quad\\;\\,&\\mathrm{on}\\;\\partial D,\n\\end{cases}\n\\end{align}\nrespectively. Similarly as (\\ref{linear001}) and (\\ref{con003}), we define a linear functional of $\\varphi$ as follows\n\\begin{align}\\label{linear002}\nQ[\\varphi]=\\int_{\\partial D_{1}}\\frac{\\partial v_{0}}{\\partial\\nu},\n\\end{align}\nwhere $v_{0}$ is defined by (\\ref{con005}). Denote\n\\begin{align*}\na_{11}:=\\int_{\\Omega}|\\nabla v_{1}|^{2}dx.\n\\end{align*}\nThen, it follows from the third line of (\\ref{con002}) and the decomposition (\\ref{con0033}) that\n\\begin{align*}\nC_{1}\\int_{\\partial D_{1}}\\frac{\\partial v_{1}}{\\partial\\nu}+\\int_{\\partial D_{1}}\\frac{\\partial v_{0}}{\\partial\\nu}=0.\n\\end{align*}\nRecalling the definition of $v_{1}$ and making use of integration by parts, we have\n\\begin{align}\\label{con007}\n\\nabla u=\\frac{Q[\\varphi]}{a_{11}}\\nabla v_{1}+\\nabla v_{0}.\n\\end{align}\n\n\n\\subsection{A general boundary value problem}\n\nTo obtain the asymptotic expansion for $\\nabla u$, we first consider the following general boundary value problem:\n\\begin{equation}\\label{con008}\n\\begin{cases}\n\\Delta v=0,\\quad\\;\\,&\\mathrm{in}\\;\\,\\Omega,\\\\\nv=\\psi,&\\mathrm{on}\\;\\,\\partial D_{1},\\\\\nv=0,&\\mathrm{on}\\;\\,\\partial D,\n\\end{cases}\n\\end{equation}\nwhere $\\psi\\in C^{2}(\\partial D_{1})$ is a given scalar function. Note that if $\\psi=1$ on $\\partial D_{1}$, then $v_{1}=v$. Extend $\\psi\\in C^{2}(\\partial D_{1})$ to $\\psi\\in C^{2}(\\overline{\\Omega})$ such that $\\|\\psi\\|_{C^{2}(\\overline{\\Omega\\setminus\\Omega_{R}})}\\leq C\\|\\psi\\|_{C^{2}(\\partial D_{1})}$. Construct a cutoff function $\\rho\\in C^{2}(\\overline{\\Omega})$ satisfying $0\\leq\\rho\\leq1$, $|\\nabla\\rho|\\leq C$ on $\\overline{\\Omega}$, and\n\\begin{align}\\label{con011}\n\\rho=1\\;\\,\\mathrm{on}\\;\\,\\Omega_{\\frac{3}{2}R},\\quad\\rho=0\\;\\,\\mathrm{on}\\;\\,\\overline{\\Omega}\\setminus\\Omega_{2R}.\n\\end{align}\nFor $x\\in\\Omega$, we define\n\\begin{align*}\n\\bar{v}(x)=[\\rho(x)\\psi(x',\\varepsilon+h_{1}(x'))+(1-\\rho(x))\\psi(x)]\\bar{u}(x),\n\\end{align*}\nwhere $\\bar{u}$ is defined by (\\ref{con009}). Specially,\n\\begin{align*}\n\\bar{v}(x)=\\psi(x',\\varepsilon+h_{1}(x'))\\bar{u}(x),\\quad\\;\\,\\mathrm{in}\\;\\Omega_{R}.\n\\end{align*}\nDue to (\\ref{con009}), we have\n\\begin{align}\\label{KK6}\n\\|\\bar{v}\\|_{C^{2}(\\Omega\\setminus\\Omega_{R})}\\leq C\\|\\psi\\|_{C^{2}(\\partial D_{1})}.\n\\end{align}\n\nSimilarly as in \\cite{LX2017}, we can obtain an asymptotic expansion of the gradient for problem (\\ref{con006}).\n\\begin{theorem}\\label{thm002}\nAssume as above. Let $v\\in H^{1}(\\Omega)$ be a weak solution of (\\ref{con008}). Then, for a sufficiently small $\\varepsilon>0$,\n\\begin{align}\\label{con013}\n|\\nabla(v-\\bar{v})(x)|\\leq C\\delta^{1-\\frac{2}{m}}(|\\psi(x',\\varepsilon+h_{1}(x'))|+\\delta^{\\frac{1}{m}}\\|\\psi\\|_{C^{2}(\\partial D_{1})}),\\quad\\mathrm{in}\\;\\,\\Omega_{R}.\n\\end{align}\nConsequently, (\\ref{con013}), together with choosing $\\psi=1$ on $\\partial D_{1}$, yields that\n\\begin{align}\\label{con015}\n\\nabla v_{1}=\\nabla\\bar{u}+O(\\mathbf{1})\\delta^{1-\\frac{2}{m}},\\quad\\;\\,\\mathrm{in}\\;\\Omega_{R},\n\\end{align}\nand\n\\begin{align*}\n\\|\\nabla v\\|_{L^{\\infty}(\\Omega\\setminus\\Omega_{R})}\\leq C\\|\\psi\\|_{C^{2}(\\partial D_{1})}.\n\\end{align*}\nwhere $v_{1}\\in H^{1}(\\Omega)$ is a weak solution of (\\ref{con006})\n\n\\end{theorem}\nNote that when $m>2$, the remainder of order $O(1)$ in \\cite{LX2017} is improved to that of order $O(\\varepsilon^{1-2\/m})$ for $x\\in\\{x'=0'\\}\\cap\\Omega_{R}$ here. For readers' convenience, the detailed proof of Theorem \\ref{thm002} is left in the Appendix. Similarly, by applying Theorem \\ref{thm002}, we can find that the leading term of $\\nabla v_{0}$ is $\\nabla\\bar{u}_{0}$ in the following.\n\\begin{lemma}\\label{lem001}\nAssume as above. Let $v_{0}$ be the weak solution of (\\ref{con005}). Then, for a sufficiently small $\\varepsilon>0$,\n\\begin{align}\\label{con018}\n\\nabla v_{0}=\\nabla\\bar{u}_{0}+O(\\mathbf{1})\\delta^{1-\\frac{2}{m}}(|\\varphi(x',h(x'))|+\\delta^{\\frac{1}{m}}\\|\\varphi\\|_{C^{2}(\\partial D)}),\\quad\\;\\,\\mathrm{in}\\;\\Omega_{R},\n\\end{align}\nand\n\\begin{align}\\label{con01818}\n\\|\\nabla_{x'}v_{0}\\|_{L^{\\infty}(\\Omega_{R})}\\leq C\\|\\varphi\\|_{C^{2}(\\partial D)},\\;\\,\\|\\nabla v_{0}\\|_{L^{\\infty}(\\Omega\\setminus\\Omega_{R})}\\leq C\\|\\varphi\\|_{C^{2}(\\partial D)},\n\\end{align}\nwhere $\\bar{u}_{0}$ is defined by (\\ref{con016}).\n\\end{lemma}\n\nTherefore, recalling the decomposition (\\ref{con007}) and in view of (\\ref{con015}) and (\\ref{con018}), for the purpose of deriving the asymptotic of $\\nabla u$, it suffices to establish the following two aspects of expansions:\n\n(i) Expansion of $Q[\\varphi]$;\n\n(ii) Expansion of $a_{11}$.\n\n\n\n\n\n\n\n\n\\section{Proofs of Theorem \\ref{thm001} and Theorem \\ref{coro002}}\n\n\\subsection{Expansion of $Q[\\varphi]$}\nBefore proving Theorem \\ref{thm001} and Theorem \\ref{coro002}, we first give an expansion of $Q[\\varphi]$ with respect to $\\varepsilon$.\n\\begin{lemma}\\label{lem002}\nAssume as above. Then, for a sufficiently small $\\varepsilon>0$,\n\n$(a)$ if ({\\bf{\\em S1}}) holds for $m\\geq n+k-1$ in Theorem \\ref{thm001},\n\\begin{align*}\nQ[\\varphi]=&\\frac{(n-1)\\omega_{n-1}\\eta\\Gamma^{n+k}_{m}}{m\\lambda^{\\frac{n+k-1}{m}}}\\rho_{k}(n,m;\\varepsilon)\n\\begin{cases}\n1+O(1)\\varepsilon^{\\frac{1}{m}},&m>n+k,\\\\\n1+O(1)\\varepsilon^{\\frac{1}{m}}|\\ln\\varepsilon|,&m=n+k,\\\\\n1+O(1)|\\ln\\varepsilon|^{-1},&m=n+k-1;\n\\end{cases}\n\\end{align*}\n\n$(b)$ if ({\\bf{\\em S1}}) holds for $mn+k,\\\\\n\\varepsilon^{-\\frac{1}{m}}+O(1)|\\ln\\varepsilon|\\|\\varphi\\|_{C^{2}(\\partial D)},&m=n+k,\\\\\n|\\ln\\varepsilon|+O(1)\\|\\varphi\\|_{C^{2}(\\partial D)},&m=n+k-1.\n\\end{cases}\n\\end{align*}\n\n{\\bf Step 2.} Proofs of $(b)$ and $(c)$. In view of the definitions of $Q[\\varphi]$ and $Q^{\\ast}[\\varphi]$, it follows from integration by parts that\n\\begin{align*}\nQ[\\varphi]=\\int_{\\partial D}\\frac{\\partial v_{1}}{\\partial\\nu}\\varphi(x),\\quad\\quad Q^{\\ast}[\\varphi]=\\int_{\\partial D}\\frac{\\partial v_{1}^{\\ast}}{\\partial\\nu}\\varphi(x),\n\\end{align*}\nwhere $v_{1}$ and $v_{1}^{\\ast}$ are defined by (\\ref{con006}) and (\\ref{con022}). Thus,\n\\begin{align*}\nQ[\\varphi]-Q^{\\ast}[\\varphi]=\\int_{\\partial D}\\frac{\\partial(v_{1}-v_{1}^{\\ast})}{\\partial\\nu}\\cdot\\varphi(x).\n\\end{align*}\n\nTo estimate $v_{1}-v_{1}^{\\ast}$, we first introduce a scar auxiliary functions $\\bar{u}^{\\ast}$ satisfying $\\bar{u}^{\\ast}=1$ on $\\partial D_{1}^{\\ast}\\setminus\\{0\\}$, $\\bar{u}^{\\ast}=0$ on $\\partial D$, and\n$$\\bar{u}^{\\ast}=\\frac{x_{n}-h(x')}{h_{1}(x')-h(x')},\\quad\\mathrm{in}\\;\\,\\Omega_{2R}^{\\ast},\\quad\\;\\,\\|\\bar{u}^{\\ast}\\|_{C^{2}(\\Omega^{\\ast}\\setminus\\Omega_{R}^{\\ast})}\\leq C,$$\nwhere $\\Omega^{\\ast}_{r}:=\\Omega^{\\ast}\\cap\\{|x'|0$,\n\n$(i)$ for $m\\geq n-1$,\n\\begin{align*}\na_{11}=&\n\\frac{(n-1)\\omega_{n-1}\\Gamma^{n}_{m}}{m\\lambda^{\\frac{n-1}{m}}}\\rho_{0}(n,m;\\varepsilon)\n\\begin{cases}\n1+O(1)\\varepsilon^{\\frac{1}{m}},&m>n,\\\\\n1+O(1)\\varepsilon^{\\frac{1}{m}}|\\ln\\varepsilon|,&m=n,\\\\\n1+O(1)|\\ln\\varepsilon|^{-1},&m=n-1;\n\\end{cases}\n\\end{align*}\n\n$(ii)$ for $m\\frac{n-1}{2},\\\\\n\\varepsilon|\\ln\\varepsilon|,&m=\\frac{n-1}{2},\\\\\n\\varepsilon,&m<\\frac{n-1}{2}.\n\\end{cases}\n\\end{align}\nBy picking $\\gamma=\\frac{1}{2m}$ in {\\bf Step 2.1} of the proof of Lemma \\ref{lem002}, it follows from (\\ref{con029})--(\\ref{con031}) and the maximum principle that\n\\begin{align*}\n|v_{1}-v_{1}^{\\ast}|\\leq C\\varepsilon^{\\frac{1}{2}},\\quad\\;\\,\\mathrm{in}\\;\\,D\\setminus\\big(\\overline{D_{1}\\cup D_{1}^{\\ast}\\cup\\mathcal{C}_{\\varepsilon^{\\frac{1}{2m}}}}\\big).\n\\end{align*}\nSimilarly as before, utilizing the standard interior and boundary estimates, we derive that\n\\begin{align}\\label{con035}\n|\\nabla(v_{1}-v_{1}^{\\ast})|\\leq C\\varepsilon^{\\frac{1}{6}},\\quad\\;\\,\\mathrm{in}\\;\\,D\\setminus\\big(\\overline{D_{1}\\cup D_{1}^{\\ast}\\cup\\mathcal{C}_{\\varepsilon^{\\frac{1}{3m}}}}\\big).\n\\end{align}\nThen combining (\\ref{con027}) and (\\ref{con035}), we obtain that\n\\begin{align}\\label{con036}\n|\\mathrm{II}_{2}|\\leq&C\\varepsilon^{\\frac{1}{6}}.\n\\end{align}\n\nAs for $\\mathrm{II}_{3}$, it follows from (\\ref{con026}) and (\\ref{con027}) that\n\\begin{align*}\n\\mathrm{II}_{3}=&\\int_{\\Omega^{\\ast}_{R}\\setminus\\Omega^{\\ast}_{\\varepsilon^{\\bar{\\gamma}}}}|\\nabla\\bar{u}^{\\ast}|^{2}+2\\int_{\\Omega^{\\ast}_{R}\\setminus\\Omega^{\\ast}_{\\varepsilon^{\\bar{\\gamma}}}}\\nabla\\bar{u}^{\\ast}\\cdot\\nabla(v_{1}^{\\ast}-\\bar{u}^{\\ast})+\\int_{\\Omega^{\\ast}_{R}\\setminus\\Omega^{\\ast}_{\\varepsilon^{\\bar{\\gamma}}}}|\\nabla(v_{1}^{\\ast}-\\bar{u}^{\\ast})|^{2}\\\\\n=&\\int_{\\varepsilon^{\\bar{\\gamma}}<|x'|n,\\\\\n\\varepsilon^{-\\frac{1}{m}}+O(1)|\\ln\\varepsilon|,&m=n,\\\\\n|\\ln\\varepsilon|+O(1),&m=n-1;\n\\end{cases}\n\\end{align*}\n\n(ii) For $m0$,\n\\begin{align}\\label{ADE009}\nF(t_{0})\\leq C\\varepsilon^{n+2-\\frac{4}{m}}\\left(|\\psi(z',\\varepsilon+h_{1}(z'))|^{2}+\\varepsilon^{\\frac{2}{m}}\\|\\psi\\|^{2}_{C^{2}}\\right).\n\\end{align}\n\n{\\bf Case 2.} If $\\varepsilon^{\\frac{1}{m}}\\leq|z'|\\leq R$ and $0 0.8$, were placed in the ETG bin. The\nclassifier algorithm in \\citet{Huertas-Company:2011} assigned a probability to\neach galaxy for belonging to four different morphological types: E\n(elliptical); S0 (lenticular); Sab (early LTG); and Scd (late LTG).\n\\citet[][in their Section 6]{Sahu:2019:II} argued that $p(\\rm E-S0) > 0.8$\nimplied that at least 10 per cent of the SDSS-DR7 ETGs selected by\n\\citetalias{Shankar:2016} using the above criteria were misclassified and may\nbe LTGs (Sab or Scd). \\citet{Sahu:2019:II} noted that the apparent offset of\nthe SMBH sample from the SDSS-DR7 ETG $\\sigma_{\\rm HL}$--$M_{\\rm *,gal}$ curve\nseems higher at the low-mass end ($10 \\lesssim \\log(M_{\\rm *,gal}\/{\\rm\n M}_{\\odot}) \\lesssim 10.5$) than at the high masses due to at least 10 per cent\ncontamination by LTGs in the supposed ETG sample of \\citetalias{Shankar:2016}.\nThis is because the LTGs (spiral) define a different \n$\\sigma_{\\rm HL}$--$M_{\\rm *,gal}$ relation than ETGs and reside below ETGs in the\n$\\sigma_{\\rm HL}$--$M_{\\rm *,gal}$ diagram \\citep[see Figure~15\n in][]{Sahu:2019:II}. This may be a reason behind a fraction of the offset \nbetween the ETG SMBH sample and the SDSS-DR7 ETG sample. However,\n\\citet{Sahu:2019:II} did not thoroughly investigate this offset which is now\ndone here.\\footnote{In 2019, we were aware of the results presented herein; \n however, pandemic-related delays have meant that we are only reporting them\n now.}\n\nImportantly, from the SDSS-DR7 spectroscopic galaxy sample without a\ndirect-dynamical SMBH mass measurement, \\citetalias{Shankar:2016} used the\nETGs within the redshift range of $0.05