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+{"text":"\\section{Introduction}\n\n\nIn this work we investigate the regularity of $p$-orthotropic functions in the plane for $1<p<2$.\nLet $\\Omega\\subset\\mathbb{R}^2$ be an open set. A weak solution of the orthotropic $p$-Laplace equation (also known as pseudo $p$-Laplace equation) is a function $u\\in W^{1,p}(\\Omega)$ such that\n\\begin{equation}\\label{orthodeg}\n\\sum_{i=1}^2 \\int_\\Omega |\\partial_i u|^{p-2} \\partial_i u \\, \\partial_i \\phi \\dx=0 \\quad \\text{for all} \\quad \\phi\\in W_0^{1,p}(\\Omega).\n\\end{equation}\nEquation \\eqref{orthodeg} arises as the Euler-Lagrange equation for the functional\n\\begin{equation}\nI_{\\Omega}(v) = \\sum_{i=1}^2\\int_{\\Omega} \\frac{|\\partial_i v|^p}{p}\\dx.\n\\end{equation}\nThe equation is singular when either one of the derivatives vanishes, and does not fall into the category of equations with $p$-Laplacian structure. \nIt was proved by Bousquet and Brasco in \\cite{BB} that weak solutions of \\eqref{orthodeg} for $1<p<\\infty$ are $C^1(\\Omega)$. A simple proof which gives a logarithmic modulus of continuity for the derivatives is contained in \\cite{LR} for the case $p\\geq 2$. The latter relies on a lemma on the oscillation of monotone functions due to Lebesgue \\cite{L} and the fact that derivatives of solutions are monotone (in the sense of Lebesgue). The purpose of this work is to extend this result to the case $1<p<2$ employing methods developed in \\cite{LR}. We obtain the following:\n\\begin{theorem}\\label{C1}\nLet $\\Omega\\subset\\mathbb{R}^2$ and $u\\in W^{1,p}(\\Omega)$ be a solution of the  equation \\eqref{orthodeg} for $1<p<2$. Fix a ball  $B_{R}\\subset\\subset \\Omega$. Then,    \n for all $j\\in\\{1,2\\}$ and $B_r\\subset\\subset B_{R\/2}$, we have\n\\begin{equation}\\label{oscEst2}\n\\osc{B_r} (\\partial_ju)\\leq C_p \n\\left(\\log\\left(\\frac{R}{r}\\right)\\right)^{-\\frac{1}{2}}\\left( \\fint_{B_{R}}  |\\nabla u|^p \\dx\\right)^\\frac{1}{p},\n\\end{equation}\nwhere $C_p$ is a constant depending only on $p$.\n\\end{theorem}\n\n\n\n\n\\paragraph{\\bf Notation.}\nWe indicate balls by $B_r=B_r(a)=\\{x\\in\\mathbb{R}^2\\;:\\; |x-a|<r\\}$ and we omit the center when not relevant. Whenever two balls $B_r\\subset B_R$ appear in a statement they are implicitly assumed to be concentric. The variable $x$ denotes the vector $(x_1, x_2)$ and we denote the partial derivatives of a function $f$ with respect to $x_j$ as $\\partial_j f$.\n\n\n\n\\section{Regularization}\nWe will consider a regularized problem by introducing a non degeneracy parameter $\\epsilon>0$. \n\\\\\n\nFix $B_R\\subset\\subset \\Omega\\subset \\mathbb{R}^2$ and  consider  the regularized Dirichlet problem\n\\begin{equation}\\label{orthonondeg2}\n\\begin{split}\n\\begin{cases}\n\\sum_{i=1}^2 \\int_{B_R} (|\\partial_i u^\\epsilon|^2 +\\epsilon)^\\frac{p-2}{2} \\partial_i u^\\epsilon\\, \\partial_i \\phi \\dx=0\\\\\nu^\\epsilon-u \\in W_0^{1,p}(B_R).\n\\end{cases}\n\\end{split}\n\\end{equation}\nNote that $u^\\epsilon$ is the unique minimizer of the regularized functional\n\\begin{equation}\nI^\\epsilon_{B_R}(v)=\\sum_{i=1}^2\\int_{B_R}\\frac{1}{p}(|\\partial_i v|^2+\\epsilon)^\\frac{p}{2}\\dx\n\\end{equation}\namong $W^{1,p}(B_R)$ functions $v$ such that $v-u\\in W^{l,p}_0(B_R)$.\nBy elliptic regularity theory, the unique solution $u^\\epsilon$ of \\eqref{orthonondeg2} is smooth in $B_R$.\\\\\nFix an index $j\\in\\{1,2\\}$. Then, replacing $\\phi$ by $\\partial_j\\phi$ in equation \\eqref{orthonondeg2} and integrating by parts, we find that the derivative $\\partial_j u^\\epsilon$ satisfies the following equation\n\\begin{equation}\\label{orthoder2}\n\\sum_{i=1}^2 \\int_{B_R} (\\epsilon+|\\partial_i u^\\epsilon|^2)^\\frac{p-4}{2} (\\epsilon+(p-1)|\\partial_i u^\\epsilon|^2)\\,  \\partial_i \\partial_j u^\\epsilon \\,\\partial_i\\phi \\dx =0\n\\end{equation}\nfor all $\\phi\\in C_0^\\infty(B_R)$.\\\\\n\n\nWe now collect some uniform estimates and convergences (see also \\cite{BB}).\n\n\\begin{Lemma}\nLet $u\\in W^{1,p}(\\Omega)$ be a solution of \\eqref{orthodeg} and $u^\\epsilon$ be a solution of \\eqref{orthonondeg2} for $1<p<2$. Then we have\n\\begin{equation}\\label{energy2}\n\\int_{B_R} |\\nabla u^\\epsilon|^p\\dx \\leq \nC_p \\left(  \\int_{B_{R}} |\\nabla u|^p\\dx +\\epsilon^\\frac{p}{2}R^2 \\right)\n\\end{equation}\nwhere $C_p$ is a constant depending only on $p$.\n\\end{Lemma}\n\n\\begin{proof}\nThe estimate follows from $$I^\\epsilon_{B_R}(u^\\epsilon)\\leq I^\\epsilon_{B_R}(u).$$\n\\end{proof}\n\n\n\\begin{Prop}\nLet $u\\in W^{1,p}(\\Omega)$ be a solution of \\eqref{orthodeg} and $u^\\epsilon$ be a solution of \\eqref{orthonondeg2} for $1<p<2$. Then, for all $j\\in\\{1,2\\}$, we have\n\\begin{align}\n\\sup_{B_{R\/2}} (\\epsilon+|\\nabla u^\\epsilon|^2)&\\leq C_p\\left( \\fint_{B_R}(\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{p}{2}\\dx \\right)^\\frac{2}{p}, \\label{lip2} \\\\ \n\\int_{B_{R\/2}}|\\nabla \\partial_j u^\\epsilon|^2\\dx &\\leq C_p\\left(\\fint_{B_R}(|\\nabla u|^p+\\epsilon^\\frac{p}{2})\\dx\\right)^\\frac{2}{p} \\label{grad2}\n\\end{align}\nwhere $C_p$ is a constant depending only on $p$.\n\\end{Prop}\n\n\\begin{proof}\nThe proof of the Lipschitz bound can be found in \\cite{FF}  while \\eqref{grad2} appears in \\cite{BB}. We provide details for completeness.\nNote that by a change of variables, the function $u^\\epsilon_R(x)=u^\\epsilon(x_0+Rx)$ satisfies the equation \n\\begin{equation}\\label{orthonondeg2scaled}\n\\sum_{i=1}^2\\int_{B_1}(|\\partial_iu^\\epsilon_R|^2+R^2\\epsilon)^\\frac{p-2}{2}\\partial_iu^\\epsilon_R\\partial_i\\phi\\dx=0 \\quad \\text{for all} \\quad \\phi\\in W_0^{1,p}(B_1).\n\\end{equation}\nIntroduce the notation $w=\\epsilon R^2+|\\nabla u^\\epsilon_R|^2$ and $a_i(z)=a_i(z_i)=(\\epsilon R^2+|z_i|^2)^\\frac{p-2}{2}z_i$ so that equation \\eqref{orthonondeg2scaled} rewrites as $$\\sum_{i=1}^2\\int_{B_1}a_i(\\partial_iu^\\epsilon_R)\\partial_i\\phi\\dx=0 \\quad \\text{for all}\\quad \\phi\\in W_0^{1,p}(B_1).$$ \nFor $j\\in\\{1,2\\}$ and $\\alpha\\geq 0$ take $\\phi=\\partial_j(\\partial_ju^\\epsilon_R\\,w^\\frac{\\alpha}{2}\\xi^2)$ so that\n$\\partial_i\\phi=\\partial_j(\\partial_i\\partial_ju^\\epsilon_R w^\\frac{\\alpha}{2} \\xi^2+\\frac{\\alpha}{2}\\partial_iw \\, w^\\frac{\\alpha-2}{2}\\,\\partial_ju^\\epsilon_R\\, \\xi^2) + 2\\partial_j(\\xi\\partial_i\\xi\\, w^\\frac{\\alpha}{2}\\,\\partial_ju^\\epsilon_R)$. Sum in $j$ to get\n\\begin{equation*}\n\\begin{split}\nA+B:&= \\sum_{i,j=1}^2\\int_{B_1} a_i(\\partial_iu^\\epsilon_R)\\partial_j(\\partial_i\\partial_ju^\\epsilon_R w^\\frac{\\alpha}{2} \\xi^2+\\frac{\\alpha}{2}\\partial_iw \\, w^\\frac{\\alpha-2}{2}\\,\\partial_ju^\\epsilon_R\\, \\xi^2)\\dx \\\\\n&+2\\sum_{i,j=1}^2\\int_{B_1}  a_i(\\partial_iu^\\epsilon_R) \\partial_j(\\xi\\partial_i\\xi\\, w^\\frac{\\alpha}{2}\\,\\partial_ju^\\epsilon_R)\\dx=0.\n\\end{split}\n\\end{equation*}\nNote that $\\partial_i w=2\\sum_{j=1}^2\\partial_i\\partial_ju^\\epsilon_R\\,\\partial_ju^\\epsilon_R$ and $\\partial_ia_i(\\partial_iu^\\epsilon_R)\\geq c_pw^\\frac{p-2}{2}$ since $1<p<2$. Integrate by parts in $A$. We get $A=A_1+A_2$ where\n\\begin{equation*}\n\\begin{split}\nA_1&:=\\sum_{i,j=1}^2\\int_{B_1}\\partial_ia_i(\\partial_iu^\\epsilon_R)(\\partial_i\\partial_ju^\\epsilon_R)^2\\,w^\\frac{\\alpha}{2}\\,\\xi^2\\dx\n\\geq c_p\\sum_{j=1}^2 \\int_{B_1} w^\\frac{p-2+\\alpha}{2}|\\nabla\\partial_ju^\\epsilon_R|^2\\xi^2\\dx,\\\\\nA_2&:=c\\alpha\\sum_{i,j=1}^2\\int_{B_1}\\partial_ia_i(\\partial_iu^\\epsilon_R)\\partial_i\\partial_ju^\\epsilon_R\\,\\partial_ju^\\epsilon _R\\,\\partial_iw\\, w^\\frac{\\alpha-2}{2}\\,\\xi^2\\dx\n=c\\alpha\\sum_{i=1}^2 \\int_{B_1}\\partial_ia_i(\\partial_iu^\\epsilon_R) (\\partial_i w)^2 w^\\frac{\\alpha-2}{2}\\,\\xi^2\\dx\\\\\n&\\geq c_p\\alpha \\int_{B_1}w^\\frac{p-4+\\alpha}{2}|\\nabla w|^2\\xi^2\\dx.\n\\end{split}\n\\end{equation*}\nNow we estimate $B=B_1+B_2+B_3$. \n\\begin{equation*}\n\\begin{split}\nB_1:&=\\sum_{i,j=1}^2\\int_{B_1}a_i(\\partial_iu^\\epsilon_R)w^\\frac{\\alpha}{2}|\\partial_ju^\\epsilon_R|\\,|\\partial_j(\\xi\\partial_i\\xi)|\\dx\n\\leq C_p\\int_{B_1}w^\\frac{p+\\alpha}{2} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\dx,\\\\\nB_2:&=\\frac{\\alpha}{2}\\sum_{i,j=1}^2\\int_{B_1} a_i(\\partial_iu^\\epsilon_R)w^\\frac{\\alpha-2}{2}|\\partial_jw|\\,|\\partial_ju^\\epsilon_R| \\,\\xi\\,|\\partial_i\\xi|\\dx \n\\leq C\\alpha\\int_{B_1} w^\\frac{p+\\alpha-2}{2}|\\nabla w|\\,\\xi\\,|\\nabla\\xi|\\dx \\\\\n&\\leq \\eta  \\alpha \\int_{B_1}w^\\frac{p-4+\\alpha}{2}|\\nabla w|^2\\xi^2\\dx+\\frac{C \\alpha}{\\eta}\\int_{B_1}|\\nabla\\xi|^2\\,w^\\frac{p+\\alpha}{2}\\dx,\\\\\nB_3:&=\\sum_{i,j=1}^2\\int_{B_1}a_i(\\partial_iu^\\epsilon_R)w^\\frac{\\alpha}{2}|\\partial_j\\partial_ju^\\epsilon_R|\\,|\\partial_ju^\\epsilon_R| \\,\\xi\\,|\\partial_i\\xi|\\dx \n\\leq \\sum_{j=1}^2\\int_{B_1}w^\\frac{p-1+\\alpha}{2}|\\nabla\\partial_ju^\\epsilon_R|\\,\\xi\\,|\\nabla\\xi|\\dx\\\\\n&\\leq \\eta\\sum_{j=1}^2 \\int_{B_1} w^\\frac{p-2+\\alpha}{2}|\\nabla\\partial_ju^\\epsilon_R|^2\\xi^2\\dx+\\frac{C}{\\eta}\\int_{B_1}|\\nabla\\xi|^2\\, w^\\frac{p+\\alpha}{2}\\dx\n\\end{split}\n\\end{equation*}\nwhere we used $a_i(\\partial_iu^\\epsilon_R)\\leq w^\\frac{p-1}{2}$ and Young's inequality with a parameter $\\eta$ to be chosen suitably small. We get\n\\begin{equation}\\label{mixEst2}\nc_p\\sum_{j=1}^2 \\int_{B_1} w^\\frac{p-2+\\alpha}{2}|\\nabla\\partial_ju^\\epsilon_R|^2\\xi^2\\dx\n+ c_p\\alpha \\int_{B_1}w^\\frac{p-4+\\alpha}{2}|\\nabla w|^2\\xi^2\\dx\n\\leq C_p(\\alpha+1)\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p+\\alpha}{2}\\dx.\n\\end{equation}\nNote that for $\\alpha=0$ we get for all $j\\in\\{1,2\\}$\n\\begin{equation}\\label{gradientEst2}\n\\int_{B_1} w^\\frac{p-2}{2}|\\nabla\\partial_ju^\\epsilon_R|^2\\xi^2\\dx\n\\leq C_p\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p}{2}\\dx,\n\\end{equation}\nand since $|\\nabla w|^2\\leq c\\sum_j |\\nabla \\partial_j u^\\epsilon_R|^2|\\nabla u^\\epsilon_R|^2$ we have\n\\begin{equation}\\label{alpha0}\n\\begin{split}\n\\int_{B_1}w^\\frac{p-4}{2}|\\nabla w|^2\\xi^2\\dx\n&\\leq c \\sum_{j=1}^2\\int_{B_1}w^\\frac{p-4}{2}|\\nabla u^\\epsilon_R|^2|\\nabla \\partial_j u^\\epsilon|^2\\xi^2\\dx\n\\leq c \\sum_{j=1}^2\\int_{B_1}w^\\frac{p-2}{2}|\\nabla \\partial_j u^\\epsilon_R|^2\\xi^2\\\\\n&\\leq C_p\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p}{2}\\dx.\n\\end{split}\n\\end{equation}\nNow for $\\alpha\\geq 1$, \\eqref{mixEst2} implies\n\\begin{equation}\n \\int_{B_1}w^\\frac{p-4+\\alpha}{2}|\\nabla w|^2\\xi^2\\dx\n\\leq C_p \\frac{\\alpha+1}{\\alpha}\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p+\\alpha}{2}\\dx\n\\end{equation}\nand combining with \\eqref{alpha0} we get\n\\begin{equation*}\n \\int_{B_1}\\lvert\\nabla (w^\\frac{p+\\alpha}{4}\\xi)\\rvert^2\\dx\n\\leq C (p+\\alpha)^2\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p+\\alpha}{2}\\dx\n\\end{equation*}\nfor all $\\alpha\\geq 0$.\nUsing Sobolev's embedding $W_0^{1,2}(B_1)\\hookrightarrow L^{2q}(B_1)$ for a fixed $q>1$ we get\n\\begin{equation}\\label{moser2}\n \\left(\\int_{B_1} w^{q\\frac{p+\\alpha}{2}}\\xi^{2q}\\dx\\right)^\\frac{1}{q}\n\\leq C_p (p+\\alpha)^2\\int_{B_1} (|\\nabla\\xi|^2+|\\nabla^2\\xi|)\\, w^\\frac{p+\\alpha}{2}\\dx.\n\\end{equation} \nNow choose a sequence of radii $r_i=1\/2^i+(1-1\/2^{i})\\frac{1}{2}$, cut-off functions $\\xi$ between $r_i$ and $r_{i+1}$ and $\\alpha_i=q^ip-p$ so that $\\frac{p+\\alpha_i}{2}=\\frac{p}{2}q^i$. Using these in \\eqref{moser2}, raising to the power $1\/q^{i}$ and iterating  we get for all $i\\in\\mathbb{N}$\n\\begin{equation*}\n\\left(\\int_{B_{r_{i+1}}} w^{\\frac{p}{2}q^{i+1}}\\dx \\right)^\\frac{1}{q^{i+1}}\n\\leq (C_p q^{2i} 2^i)^\\frac{1}{q^i}\\left( \\int_{B_{r_i}} w^{\\frac{p}{2}q^i}\\dx\\right)^\\frac{1}{q^i}\n\\leq \\prod_{j=0}^i(C_p q^{2j} 2^j)^\\frac{1}{q^j} \\int_{B_{1}} w^{\\frac{p}{2}}\\dx.\n\\end{equation*} \nObserve that $\\prod_{i=0}^\\infty(C_p q^{2i} 2^i)^\\frac{1}{q^i}=C(p,q)<\\infty$ so passing to the limit as $i\\to\\infty$ we get\n$$\\sup_{B_{1\/2}}w^\\frac{p}{2}\\leq C(p,q)\\int_{B_{1}} w^{\\frac{p}{2}}\\dx$$\nwhich, after rescaling, proves \\eqref{lip2}. Now going back to \\eqref{gradientEst2}, choosing a cut-off function between $B_{R\/2}$ and $B_R$ and using $1<p<2$ we get\n\\begin{equation*}\n\\int_{B_{R\/2}} |\\nabla \\partial_j u^\\epsilon|^2\\dx \\leq \nC_p\\sup_{B_{R\/2}} (\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{2-p}{p} \\fint_{B_R}(\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{p}{2}\\dx.\n\\end{equation*}\nUsing \\eqref{lip2} and \\eqref{energy2} we obtain \\eqref{grad2}.\n\\end{proof}\n\nNext we collect some facts about the convergence of $u^\\epsilon$ to the solution of the degenerate equation. These are sufficient for our purposes.\n\\begin{Prop}\\label{uni2}\nLet $u^\\epsilon$ be the solution of \\eqref{orthonondeg2} for $1<p<2$ and $u\\in W^{1,p}(\\Omega)$ the solution of \\eqref{orthodeg}. We have\n\\begin{itemize}\n\\item $u^\\epsilon$ converges to $u$ locally uniformly in $B_R$,\\\\\n\\item $\\nabla u^\\epsilon$ converges to $\\nabla u$ in $L^p(B_R)$.\n\\end{itemize}\n\\end{Prop}\n\n\\begin{proof}\nFrom the energy estimate \\eqref{energy2} we obtain a uniform bound for the $L^p$ norm of $\\nabla u^\\epsilon$. Therefore (up to a subsequence) $u^\\epsilon$ converges to some $v\\in W^{1,p}(B_R)$ weakly in $W^{1,p}(B_R)$ and strongly in $L^p(B_R)$. Note that we have $v-u\\in W_0^{1,p}(B_R)$. By weakly lower semicontinuity we get\n\\begin{equation*}\n\\begin{split}\nI_{B_R}(v) = \\sum_{i=1}^2\\int_{B_R}\\frac{|\\partial_i v|^p}{p}\\dx\n&\\leq \\liminf_{\\epsilon\\to 0} \\sum_{i=1}^2\\int_{B_R}\\frac{|\\partial_i u^\\epsilon|^p}{p}\\dx\\\\\n&\\leq \\liminf_{\\epsilon\\to 0} \\sum_{i=1}^2\\int_{B_R}\\frac{1}{p}(|\\partial_i u^\\epsilon|^2+\\epsilon)^\\frac{p}{2}\\dx\\\\\n&\\leq \\liminf_{\\epsilon\\to 0} \\sum_{i=1}^2\\int_{B_R}\\frac{1}{p}(|\\partial_i u|^2+\\epsilon)^\\frac{p}{2}\\dx\\\\\n&=\\sum_{i=1}^2\\int_{B_R}\\frac{1}{p}|\\partial_i u|^p\\dx\n=I_{B_R}(u).\n\\end{split}\n\\end{equation*}\nNote that in the third inequality we used the minimality of $u^\\epsilon$ subject to the boundary condition $u^\\epsilon-u\\in W_0^{1,p}(B_R)$. By uniqueness of the minimizer of $I_{B_R}$ among functions with boundary values $u$ in $B_R$, we get $v=u$. \nBy the uniform Lipschitz estimate \\eqref{lip2} and Ascoli-Arzela' theorem we obtain that the convergence is uniform.\\\\\n\nNow we show $L^p(B_R)$ convergence of the gradient. Use $\\phi = u^\\epsilon-u$ as a test function in \\eqref{orthonondeg2}, add and subtract the term $(|\\partial_i u|^2+\\epsilon)^\\frac{p-2}{2}\\partial_i u$ to get\n\\begin{equation*}\n\\begin{split}\n\\sum_{i=1}^2\\int_{B_R} &\\left((|\\partial_i u^\\epsilon|^2+\\epsilon)^\\frac{p-2}{2}\\partial_i u^\\epsilon-(|\\partial_i u|^2+\\epsilon)^\\frac{p-2}{2}\\partial_i u\\right)\\left(\\partial_i u^\\epsilon-\\partial_i u\\right)\\dx\\\\\n&=\\sum_{i=1}^2\\int_{B_R} (|\\partial_i u|^2+\\epsilon)^\\frac{p-2}{2}\\partial_i u (\\partial_i u-\\partial_i u^\\epsilon)\\dx.\n\\end{split}\n\\end{equation*}\nSince  $\\partial_i u-\\partial_i u^\\epsilon$  converges to $0$ weakly in $L^p(B_R)$, the integral in the right hand side converges to $0$. We can minorize the integral in the left hand side using the inequality \n$$ |a-b|^2(\\epsilon+|a|^2+|b^2|)^\\frac{p-2}{2}\\leq C_p ((\\epsilon+|a|^2)^\\frac{p-2}{2}a-(\\epsilon+|b|^2)^\\frac{p-2}{2}b)(a-b) $$\nvalid for $1<p<2$, and obtain that\n\\begin{equation}\\label{conv0}\n\\int_{B_R}\\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p-2}{2}|\\partial_i u^\\epsilon-\\partial_i u|^2\\dx \\longrightarrow 0\n\\end{equation} \nas $\\epsilon\\to 0$, for $i=1$, $2$.\nFinally by H\\\"older's inequality\n\\begin{equation*}\n\\begin{split}\n\\int_{B_R} |\\partial_iu^\\epsilon-\\partial_iu|^p\\dx\n&= \\int_{B_R}  |\\partial_iu^\\epsilon-\\partial_iu|^p \\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p(p-2)}{2}\\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p(2-p)}{2}\\dx\\\\\n&\\leq \\left( \\int_{B_R} |\\partial_iu^\\epsilon-\\partial_iu|^2 \\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p-2}{2}\\dx\\right)^\\frac{p}{2}\\cdot\\\\\n&\\qquad\\qquad\\qquad\\quad\\cdot\\left( \\int_{B_R} \\left(\\epsilon+|\\partial_i u^\\epsilon|^2+|\\partial_i u|^2\\right)^\\frac{p}{2}\\dx\\right)^\\frac{2-p}{2}.\n\\end{split}\n\\end{equation*}\nSince the last integral is uniformly bounded in $\\epsilon$, using \\eqref{conv0} we get  that $\\partial_i u^\\epsilon$ converges to $\\partial_i u$ in $L^p(B_R)$.\n\\end{proof}\n\n\\section{Monotone functions and Lebesgue's lemma}\nA continuous function $v:\\Omega\\longrightarrow\\mathbb{R}$ is monotone (in the sense of Lebesgue) if\n$$\\max_{\\overline{D}}v=\\max_{\\partial D}v\\quad \\text{and}\\quad  \\min_{\\overline{D}}v=\\min_{\\partial D}v$$\nfor all subdomains $D\\subset\\subset \\Omega$. Monotone functions are further discussed in \\cite{Manf}.\n\nThe next Lemma is due to Lebesgue \\cite{L}.\n\\begin{Lemma}\\label{oscLeb}\nLet $B_R\\subset \\mathbb{R}^2$ and  $v\\in C(B_R )\\cap W^{1,2}(B_R)$ be monotone in the sense of Lebesgue. Then\n\\begin{equation*}\n(\\osc{ B_r} v)^2\\log\\left(\\frac{R}{r}\\right)\n\\leq  \\pi \\int_{B_R\\setminus B_r} |\\nabla v(x)|^2 \\dx\n\\end{equation*}\nfor every $r<R$.\n\\end{Lemma}\n\n\\begin{proof}\nAssume $v$ is smooth. Let $(\\eta, \\zeta)$ be the center of $B_R$. Let $x_1$ and $x_2$ be two points on the circle of radius $t$, and let $\\gamma:[0,2\\pi]\\longrightarrow \\mathbb{R}^2$, $\\gamma(s)=(\\eta+t\\cos(s),\\zeta+ t\\sin(s))$ be a parametrization of the circle such that $\\gamma(a)=x_1$ and $\\gamma(b)=x_2$. Then we have\n\\begin{equation*}\n\\begin{split}\nv(x_1)-v(x_2)=\\int_a^b \\frac{d}{ds} v(\\gamma(s)) \\ds \n= \\int_a^b \\langle  \\nabla v(\\gamma(s)), \\gamma '(s) \\rangle \\ds\n\\leq \\int_a^{b} t\\, |\\nabla v(\\gamma(s))| \\ds.\n\\end{split}\n\\end{equation*}\nTaking the supremum on angles $a$ and $b$ such that $|a-b|\\leq \\pi$ and using H\\\"older's inequality, we get\n\\begin{equation*}\n(\\osc{\\partial B_t}v )^2 \n\\leq \\pi t^2  \\int_0^{2\\pi}  |\\nabla v (\\gamma(s))|^2 \\ds.\n\\end{equation*}\nNow diving by $t$, integrating from $r$ to $R$, and using polar coordinates we get\n\\begin{equation*}\n\\int_r^R\\frac{(\\text{osc}_{\\partial B_t}v )^2}{t} \\dt\n\\leq \\pi \\int_r^R \\int_0^{2\\pi} t\\,|\\nabla v(\\gamma(s))|^2 \\ds \\dt\n= \\pi \\int_{B_R\\setminus B_r} |\\nabla v(x)|^2 \\dx.\n\\end{equation*}\nThanks to the monotonicity of $v$, for $t\\geq r$ we have\n\\begin{equation*}\n\\osc{\\partial B_t} \\partial_ju^\\epsilon \\geq\n   \\osc{B_t} \\partial_ju^\\epsilon\n\\geq \\osc{B_r} \\partial_ju^\\epsilon\n\\end{equation*}\nand we get the result for a smooth function. The general statement follows by approximation.\n\\end{proof}\n\n\nThe following is credited to \\cite{BB} (see Lemma 2.14 for the minimum principle).\n\\begin{Lemma}\\label{maxmin2}[Minimum and Maximum principles for the derivatives]\\\\\nLet $u^\\epsilon$ be the solution of \\eqref{orthonondeg2}. Then \n$$\\min_{\\partial B_r}\\partial_j u^\\epsilon\\leq \\partial_j u^\\epsilon (x)\\leq\\max_{\\partial B_r}\\partial_j u^\\epsilon $$\nfor all $x\\in B_r$, $B_r\\subset\\subset B_R$ and $j=1$, $2$. In particular, $\\partial_j u^\\epsilon$ is monotone in the sense of Lebesgue.\n\n\\end{Lemma}\n\\begin{proof}\nWe are going to show that given a constant $C$, if $\\partial_j u^\\epsilon \\leq C$ (resp. $\\partial_j u^\\epsilon \\geq C$) in $\\partial B_r$ then $\\partial_j u^\\epsilon \\leq C$ (resp. $\\partial_j u^\\epsilon \\geq C$) in $B_r$.\nLet $\\phi^{\\pm}= 1_{B_r}(\\partial_j u^\\epsilon-C)^\\pm =1_{B_r}\\max\\{\\pm(\\partial_j u^\\epsilon-C),0\\} $ in the equation satisfied by the derivative \\eqref{orthoder2}. Since $u^\\epsilon$ is smooth and $\\partial_j u^\\epsilon \\geq C$  (resp. $\\partial_j u^\\epsilon \\leq C$) on $\\partial B_r$ we have $\\phi^\\pm\\in W_0^{1,2}(\\Omega)$, so they are admissible functions. We get\n\\begin{equation*}\n\\begin{split}\n0&=\\sum_{i=1}^2 \\int_{B_r} (\\epsilon+|\\partial_i u^\\epsilon|^2)^\\frac{p-4}{2} (\\epsilon+(p-1)|\\partial_i u^\\epsilon|^2)\\, |\\partial_i (\\partial_j u^\\epsilon -C)^\\pm |^2 \\dx \\\\\n&\\geq \\epsilon \\sum_{i=1}^2\\int_{B_r}(\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{p-4}{2}  |\\partial_i (\\partial_j u^\\epsilon -C)^\\pm |^2 \\dx \\\\\n&= \\epsilon \\int_{B_r}(\\epsilon+|\\nabla u^\\epsilon|^2)^\\frac{p-4}{2}  |\\nabla (\\partial_j u^\\epsilon -C)^\\pm |^2 \\dx .\n\\end{split}\n\\end{equation*}\nThis implies $(\\partial_j u^\\epsilon -C)^\\pm$ is constant in $B_r$, and since it is $0$ in $\\partial B_r$ then $(\\partial_j u^\\epsilon -C)^\\pm=0$ in $B_r$.\n\\end{proof}\n\\\n\n\\section{Proof of the Main Theorem}\n\\begin{proof}\n[Proof of Theorem \\eqref{C1}]\nApplying Lemma \\eqref{oscLeb} and estimate \\eqref{grad2} we get for all $r<R\/2$\n\\begin{equation}\n(\\osc{B_r} \\partial_ju^\\epsilon)^2 \\log(\\frac{R}{r}) \\leq C \\norm{\\nabla \\partial_ju^\\epsilon}^2_{L^2(B_{R\/2})} \\leq C \\left(\\fint_{B_{R}} |\\nabla u|^p\\dx+\\epsilon^\\frac{p}{2}\\right)^\\frac{2}{p}\n\\end{equation}\nand hence for all $r<R\/2$\n\\begin{equation}\\label{osc2}\n\\osc{B_r} \\partial_ju^\\epsilon  \\leq C \\left(\\log\\left(\\frac{R}{r}\\right)\\right)^{-\\frac{1}{2}}\n\\left(\\fint_{B_{R}} |\\nabla u|^p\\dx+\\epsilon^\\frac{p}{2}\\right)^\\frac{1}{p}\n\\end{equation}\nwhere $C$ is a constant independent of $\\epsilon$.\n\nThanks to Proposition \\eqref{uni2} we can pass to the limit and get \\eqref{oscEst2}.\n\\end{proof}\n\n\n\\subsection{Acknowledgements}\nI thank Peter Lindqvist for useful comments and suggestions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe success of supervised deep learning approaches can be attributed to the availability of large quantities of labeled data that is essential for network training \\cite{basak2020single}. However, in the biomedical domain \\cite{shurrab2022self, ronneberger2015unet, pavlopoulos2019survey}, it is difficult to acquire such large quantities of annotated data as the annotations are to be done by trained medical professionals. Although supervised learning has been used extensively in the past decade in biomedical imaging \\cite{pavlopoulos2019survey, basak2022mfsnet}, Self-supervised Learning (SSL) \\cite{doersch2015unsupervised, pathak2016context, chen2020simple, he2020momentum, manna2022swis} provides more traction for sustaining deep learning methods in the medical vision domain \\cite{basak2022embarrassingly, chen2019self, taleb20203d}. SSL-based pre-training alleviates the data annotation problem by utilizing only unstructured data to learn distinctive information which can further be utilized in downstream applications, typically in a semi-supervised fashion \\cite{basak2022addressing, chaitanya2020contrastive, basak2022embarrassingly}. Typically, SSL have shown to be exceedingly promising in domains with enormous amounts of data such as natural images \\cite{chen2020simple, he2020momentum} with a focus on both contrastive \\cite{oord2018representation, he2020momentum, chen2020simple, wang2021dense} and non-contrastive variants \\cite{zbontar2021barlow, bardes2021vicreg}. Recently, SSL has begun to be extensively employed for medical images as well, as evidenced by the survey paper by Shurrab \\textit{et. al. } \\cite{shurrab2022self}.\n\n\\begin{figure*}[!t]\n\\begin{center}\n  \\includegraphics[width=\\linewidth]{teaser_landscape.pdf}\n\\end{center}\n\\vspace{-0.4cm}\n \\caption{Schematic representation of contrastive learning-based pre-training of (a) baseline model \\cite{chaitanya2020contrastive} (b) our IDEAL model. $\\mathcal{E}$ represents the encoder backbone; $GAP =$ Global Average Pooling; $x_q$ and $x_k$ are the query and key images respectively. The convolution projection head in the IDEAL model generates dense projections that give better local clusters than using a traditional global pooling followed by a Multi-Layer Perceptron (MLP).}\n\\vspace{-0.4cm}\n\\label{fig:teaser}\n\\end{figure*}\n\nOne of the most popular ways of employing SSL is contrastive learning \\cite{oord2018representation, he2020momentum, chen2020simple}, which owing to its prolific performance across various vision tasks has almost become a de facto standard for self-supervision. The intuition of contrastive learning is to \\textit{pull} embeddings of semantically similar objects closer and simultaneously \\textit{push} away the representations of dissimilar objects. In medical image segmentation, contrastive learning has been leveraged in a few prior works \\cite{chaitanya2020contrastive, zeng2021positional}.  However, naive contrastive learning frameworks such as SimCLR \\cite{chen2020simple} and MoCo \\cite{he2020momentum} learn a global representation and thus, are not suitable to be applied directly to segmentation, a pixel-level task. The seminal work by Chaitanya \\textit{et. al. } \\cite{chaitanya2020contrastive} utilized a contrastive pre-training strategy to learn useful representations for a segmentation task in a low-data regime. Despite having success, their method has a few limitations, which we attempt to address and improve upon in this paper. In particular, we take a cue from the aforementioned work \\cite{chaitanya2020contrastive} to propose a novel contrastive learning-based medical image segmentation framework.\n\nAlthough our work is built upon \\cite{chaitanya2020contrastive}, there are several salient points of difference between the two models that render our contributions non-trivial and unique. First of all, our proposed method preserves spatial information and constructs a dense output projection that preserves locality. This is in contrast to the projection used in \\cite{chaitanya2020contrastive} where a global pooling is applied to the backbone encoder, thereby obtaining a single global feature representation vector for every input image. In other words, \\cite{chaitanya2020contrastive}  employed global contrastive learning for encoder pre-training and the local contrastive learning is only restricted to the decoder fine-tuning. in contrast, our method involves pre-training of the encoder on dense local features. The intuitive difference between our proposed framework and that of \\cite{chaitanya2020contrastive} has been depicted in Fig. \\ref{fig:teaser}. We argue that this will benefit pixel-level informed downstream tasks such as segmentation. This statement is supported by our findings as well (refer to Section \\ref{sec:expt} for further details). \n\nSecondly, our definition of finding positive and negative pairs for contrastive learning is different from that of traditional contrastive \\cite{chen2020simple, he2020momentum} methods. This is because, in our case, we find dense correspondences across views and define positives and negatives accordingly, i.e., we perform feature-level Contrastive Learning (refer to Section \\ref{sec:method}). Further, we adopt a novel bidirectional consistency regularization mechanism that perturbs the network instead of image perturbation used by most frameworks \\cite{he2020momentum, zou2020pseudoseg, basak2022embarrassingly}. By this approach, we enforce both streams of data flow to learn better features competitively, unlike the traditional image augmentation-based consistency regularization approaches where only one model stream learns through a single backpropagation. The empirical evaluation suggests that our proposed consistency regularization yields a superior segmentation performance, outperforming existing state-of-the-art semi-supervised approaches in literature $^\\dagger$ \\cite{bai2017semi, chaitanya2019semi}. Our contributions are as follows:\n\\begin{enumerate}\n    \\item We propose an SSL strategy that leverages dense projection head representations for contrastive learning for learning robust local features. \n    \n    \\item We redefine 'positive' and 'negative' samples in contrastive learning and extend the InfoNCE loss objective to adapt to the dense representations during the pre-training phase.\n    \n    \\item Our work also employs a uniquely devised cross-consistency regularization for fine-tuning the network to the downstream segmentation task.\n\\end{enumerate}\n\n\\begin{figure*}[tbp]\n    \\centering\n    \\includegraphics[width=\\linewidth]{mainfig_landscape.pdf}\n    \\caption{An overview of the proposed IDEAL framework: (a) \\textit{Pre-training}- $x_q$ and $x_k$ are the query and key images, $\\mathcal{E}$ and $\\mathcal{G}$ represent the encoder and projection head, respectively. The projection head employs a 1x1 convolution layer instead of a traditional MLP for dense feature extraction, resulting in better local clustering of features. (b) \\textit{Fine-tuning}- Two perturbed branches with the same input are employed. $\\mathcal{E}(\\theta)$ is the shared encoder initialized similarly for both streams, $\\mathcal{D}(\\theta_1)$ and $\\mathcal{D}(\\theta_2)$ represent two different decoder architectures; $p_1$ and $p_2$ are the predicted output segmentation maps which are thresholded to obtain $y_1$ and $y_2$ respectively. $y_1$ backpropagates through the second stream and $y_2$ backpropagates through the first stream enforcing cross-consistency in segmentation.}\n    \\label{fig:mainfig}\n\\end{figure*}\n\n    \n    \n\n\n\n\n\\def$\\dagger$}\\footnotetext{\\hspace{-1.5mm}Code will be released upon acceptance}\\def\\thefootnote{\\arabic{footnote}{$\\dagger$}\\footnotetext{\\hspace{-1.5mm}Code will be released upon acceptance}\\def$\\dagger$}\\footnotetext{\\hspace{-1.5mm}Code will be released upon acceptance}\\def\\thefootnote{\\arabic{footnote}{\\arabic{footnote}}\n\\section{Methodology}\\label{sec:method}\nOur proposed method consists of two parts: (1) Self-Supervised pre-training of the encoder, and (2) Semi-Supervised fine-tuning of the network using consistency regularization. Taking reference from \\cite{chaitanya2020contrastive}, we propose a pairwise contrastive (dis)similarity learning, employing global and local feature exploration for pre-training the encoder. The pre-trained encoder is thereafter transferred for the downstream task of cross-supervision-based consistency regularization, which aims to fine-tune the network. \n\n\\subsection{Background: Global Contrastive Learning}\\label{sec:GCL}\nThe widely employed variant of self-supervised representation learning is contrastive learning \\cite{he2020momentum, chen2020simple, oord2018representation}, where the images and their perturbations are fed into a cascade of the shared encoder and projection head. The obtained features (global representations) of each of these images in the mini-batches are then utilized for a contrastive loss function, which is designed considering contrastive learning as a dictionary lookup task \\cite{wang2021dense}. Out of all the encoded keys in the set $k$ for each encoded query $q$, a single positive key $k^+$ is matched, whilst the remainder of the keys (negative keys) represent different images of the mini-batch. A contrastive loss function is represented as follows:\n\\begin{equation}\n    \\mathcal{L}_g = -\\log \\frac{\\exp(q\\cdot k^+\/\\tau)}{\\exp(q\\cdot k^+\/\\tau) + \\sum\\limits_{k^-}\\exp(q\\cdot k^-\/\\tau)}\n\\end{equation}\n\n\\subsection{Improved Local Contrastive Learning}\\label{sec:LCL}\nWe employ local contrastive Learning that extends the existing global contrastive framework into a dense paradigm. The key difference between the traditional contrastive framework and the current work lies in the formulation of the loss function and the projection head. The traditional contrastive learning follows the paradigm where features are extracted by a backbone encoder (e.g. ResNet \\cite{he2016deep}), followed by a projection head. The projection head typically consists of a global pooling operation and an MLP module comprising a few FC layers with ReLU layers in between them. However, in our case, we remove the entire global pooling and MLP part and replace it with a $1\\times1$ convolution operation ($\\mathcal{G}$). This projection head generates dense feature representations, with an equal number of parameters as the traditional ones. \n\nWe define a set of keys $\\{k_0, k_1, ....\\}$ extracted from the encoder for every query $q$, where the encoded keys represent the local part of the image, which is very different from traditional contrastive learning where every key corresponds to a single image view. These keys represent individual features of the $D_h\\times D_w$ vectors extracted from the encoder followed by $1\\times1$ convolution, where $D_h$ and $D_w$ represent the spatial dimension of the extracted feature maps. The positive keys $k^+$ are pooled from the extracted correspondence across views, i.e. one of the $D_h\\times D_w$ features extracted from different views of the same image. The negative key $k^-$ is assigned from features from a view of different images. Having this assignment strategy and taking reference from InfoNCE \\cite{oord2018representation}, we define the local contrastive loss as:\n\n\\begin{equation}\n    \\mathcal{L}_{loc} = \\frac{-1}{D_h D_w} \\sum\\limits_{i=1}^{D_h D_w} \\log \\frac{\\exp(q_i\\cdot k_i^+\/\\tau)}{\\exp(q_i\\cdot k_i^+\/\\tau) + \\sum\\limits_{k_i^-}\\exp(q_i\\cdot k_i^-\/\\tau)}\n\\end{equation}\n\n\\subsection{Semi-Supervised Fine-tuning}\nThe SSL pre-training phase is followed by a semi-supervised fine-tuning of the trained encoder network for our downstream segmentation task. Consistency regularization \\cite{basak2022embarrassingly, chaitanya2020contrastive}, a very popular semi-supervised approach for segmentation, has been employed here, however, with two very unique and non-trivial customization, which are discussed below. \n\nIn typical consistency regularization schemes, such as in MoCo \\cite{he2020momentum}, dampened sharing of weights takes place i.e. one branch is learned directly following the gradient updates whereas the other branch is learned by employing an Exponential Moving Average (EMA) weight transfer strategy. Such a learning paradigm limits the feature learning ability of the second branch. To alleviate this, we consider two decoders of different architecture initialization. Furthermore, the up-scaling convolutions corresponding to each of them are made to work differently, i.e., one of them has ConvTranspose (convolution with trainable kernels) whilst the other one has an Up-sampling layer (simple bilinear interpolation). Outputs of each of these decoders are passed through sigmoid activation functions resulting in $p_i$ which are then threshold-ed to generate one-hot $y_i$ (see Fig. \\ref{fig:mainfig}). For labeled samples, we compute a simple cross-entropy loss between target mask $m$ and the predicted outputs $p_i$. Otherwise, a cross-consistency loss is enforced between the output probabilities $p_i$ from the two decoder branches, guided by the one-hot pseudo-label $y_j$ coming from the other branch. In other words, $y_1$ acts as a pseudo-supervisory signal for $p_2$, and $y_2$ for $p_1$. Thus, one branch learns based on the other branch's output, leading to a \\emph{competitive learning} between two branches, resulting more confident predictions. The loss function is defined as:\n\n\\begin{equation}\n\\mathcal{L}_{SSF} =\n\\begin{cases}\nCE(p_1, m) + CE(p_2, m),     \\text{ for labeled set}\\\\\nCE(p_1, y_2) + CE(p_2, y_1), \\text{ for unlabeled set}\n\\end{cases}\n\\end{equation}\n\n\n\n\n\n\\section{Experiments and Results}\\label{sec:expt}\n\\subsection{Experimental Setup}\n\\keypoint{Datasets:}\\label{sec:data} The proposed method has been evaluated on two publicly available MRI datasets - (1) the ACDC dataset \\cite{bernard2018deep}, consisting of 100 cardiac 3D shot-axis MRI volumes with expert manual annotations for three structures, namely, the left and right ventricles and the myocardium; and (2) the MMWHS dataset \\cite{zhuang2013challenges, zhuang2016multi}, consisting of 20 cardiac 3D MRI volumes with expert manual annotations for seven structures: the left and right ventricles, the left and right atria, the pulmonary artery, the myocardium, and the ascending aorta. The datasets are split into train, validation, and test sets, where the validation set for each dataset consists of 2 volumes; the test set consists of 20 volumes for the ACDC dataset and 10 volumes for the MMWHS dataset. The rest of the volumes (78 for ACDC and 8 for MMWHS) are used for training. The training and validation set together form the pre-training set, while the testing set was used \\emph{solely} for model evaluation.\n\n\n\\keypoint{Implementation Details:}\\label{sec:impl} \nIDEAL uses a ResNet-50 \\cite{he2016deep} encoder backbone with ADAM optimizer and an initial learning rate of $1e-5$, and a U-Net \\cite{ronneberger2015unet} decoder with 4 upscaling layers. The IDEAL model was implemented in PyTorch \\cite{paszke2019pytorch} and accelerated by an NVIDIA Tesla K80 GPU.\n\n\\keypoint{Evaluation Metrics:}\\label{sec:eval} Three different metrics have been used to evaluate the IDEAL model on the segmentation tasks: Dice Similarity Coefficient (DSC), Average Symmetric Distance (ASD), and Hausdorff Distance (HD). The average scores over 5 runs for all the metrics have been reported for all the fine-tuning experiments.\n\n\n\n\n\\subsection{Performance Analysis}\\label{sec:performance}\n\n\\subsubsection{Fine-tuning performances on limited annotations}\n\n\\begin{table}[!ht]\n    \\centering\n    \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{l cccc cccc}\n\n    \\toprule\n    \\textbf{Metric} & \\multicolumn{4}{c}{\\textbf{ACDC}} & \\multicolumn{4}{c}{\\textbf{MMWHS}}\\\\[0.5ex]\n    \\cmidrule(lr){2-5}\n    \\cmidrule(lr){6-9}\n    & \\textbf{L=1.25\\%} & \\textbf{L=2.5\\%} & \\textbf{L=10\\%} & \\textbf{L=100\\%} & \\textbf{L=10\\%} & \\textbf{L=20\\%} & \\textbf{L=40\\%} & \\textbf{L=100\\%} \\\\[0.5ex]\n    \\midrule\n    \n    \\textbf{ASD} $\\downarrow$  & 0.677\t& 0.614\t& 0.556\t& 0.489\t& 2.397\t& 1.798\t& 1.355\t& 1.317 \\\\[.5ex]\n    \n    \\textbf{HD} $\\downarrow$ & 2.409 & 2.209 & 2.094 & 1.999 & 5.001 & 3.499 & 2.221 & 2.183 \\\\[.5ex]\n\n    \\textbf{DSC} $\\uparrow$ & 0.738 & 0.846 & 0.879 & 0.882 & 0.626 & 0.791 & 0.815 & 0.826 \\\\[.5ex]\n    \n    \\bottomrule\n    \n    \\end{tabular} %\n    }\n    \\caption{Results obtained by the IDEAL framework with varying amounts of labeled data on the ACDC and MMWHS datasets. `$L$' represents the amount of labeled data used.}\n    \\label{tab:metrics}\n\\end{table}\n\nTo study the segmentation performance under limited labels during the fine-tuning phase, we have experimented with 1.25\\%, 2.5\\%, and 10\\% labeled volumes (denoted by `$L$' henceforth) for ACDC and 10\\%, 20\\%, and 40\\% labeled volumes for the MMWHS dataset, following the existing works in literature that have used the same datasets \\cite{basak2022embarrassingly, chaitanya2020contrastive}. The amount of labeled data is expressed as a percentage of the training set. The results are tabulated in Table \\ref{tab:metrics}. From the results, it can be inferred that on both datasets, the expected trend of improving metrics with an increase in labeled data for the fine-tuning phase. Further, it is promising to observe that even with a fraction of labels available (10\\% for ACDC and 40\\% for MMWHS), the respective semi-supervised setups asymptotically approach the fully supervised setup (differs by 0.3\\% in ACDC and 1.1\\% in MMWHS), thus depicting the efficacy of our framework under limited annotations. \n\n\n\n\n\n\\subsubsection{Comparison to state-of-the-art}\n\n\nWe have also compared our proposed framework with several existing state-of-the-art semi-supervised cardiac MRI segmentation methods in literature \\cite{chaitanya2020contrastive, chen2020simple, zeng2021positional, chen2019self, hu2021semi, chaitanya2019semi, bai2017semi, wu2022mutual}, the results of which are tabulated in Table \\ref{tab:sotas}. Additionally, we also provide a visual comparison of our segmentation performances with the SoTA methods in Fig. \\ref{fig:segoutputs}. \n\n\n\\begin{table}[t]\n    \\centering\n   \n    \\caption{Performance Comparison (DSC scores) of the proposed IDEAL framework with SoTA methods in the literature on the ACDC and MMWHS datasets.}\n    \\label{tab:sotas}\n    \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{l ccc ccc}\n\n    \\toprule\n    \\textbf{Method} & \\multicolumn{3}{c}{\\textbf{Average DSC (ACDC)}} & \\multicolumn{3}{c}{\\textbf{Average DSC (MMWHS)}}\\\\[0.5ex]\n    \\cmidrule(lr){2-4}\n    \\cmidrule(lr){5-7}\n    & \\textbf{L=1.25\\%} & \\textbf{L=2.5\\%} & \\textbf{L=10\\%} & \\textbf{L=10\\%} & \\textbf{L=20\\%} & \\textbf{L=40\\%}\\\\[0.5ex]\n    \\midrule\n    \n    Chaitanya \\textit{et. al. } \\cite{chaitanya2020contrastive} & 0.725 & 0.789 & 0.872 & 0.569 & 0.694 & 0.794 \\\\[.5ex]\n\n\n    Global CL \\cite{chen2020simple} & 0.729 & \\textbf{--} & 0.847 & 0.500 & 0.659 & 0.785 \\\\[.5ex]\n    \n    \n    PCL \\cite{zeng2021positional} & 0.671 & \\textbf{0.850} & \\textbf{0.885} & \\textbf{--} & \\textbf{--} & \\textbf{--} \\\\[.5ex]\n    \n    \n    Context Restoration \\cite{chen2019self} & 0.625 & 0.714 & 0.851 & 0.482 & 0.654 & 0.783 \\\\[.5ex]\n    \n    MC-Net \\cite{wu2022mutual} & 0.677 & 0.724 & 0.855 & 0.551 & 0.654  & 0.798 \\\\[.5ex]\n    \n    Label Efficient \\cite{hu2021semi} & \\textbf{--} & \\textbf{--} & \\textbf{--} & 0.382 & 0.553 & 0.764 \\\\[.5ex]\n    \n    \n    Data Augmentation \\cite{chaitanya2019semi} & 0.731 & 0.786 & 0.865 & 0.529 & 0.661 & 0.785 \\\\[.5ex]\n     \n     \n    Self Train \\cite{bai2017semi} & 0.690 & 0.749 & 0.860 & 0.563 & 0.691 & 0.801 \\\\[.5ex]\n    \n    \n    \\rowcolor{LightCyan}\n    Ours & \\textbf{0.738} & 0.846 & 0.879 & \\textbf{0.626} & \\textbf{0.791} & \\textbf{0.815} \\\\[.5ex]\n    \n    \\bottomrule\n    \n    \\end{tabular} %\n    }\n    \n\\end{table}\n\n\n\\keypoint{ACDC Dataset} From Table \\ref{tab:sotas}, it can be observed that our IDEAL model performs the best in $ L=1.25\\%$ setting, while being highly competitive to  PCL \\cite{zeng2021positional} in $L=2.5\\% (\\sim0.004)$ and $L=10\\% (\\sim0.006)$. This shows that even in severely constrained conditions, IDEAL can perform optimally.\n\n\\keypoint{MMWHS Dataset} Compared to existing state-of-the-art methods, our IDEAL model outperforms all previous frameworks by large margins, as inferred from Table \\ref{tab:sotas}. Chaitanya \\textit{et. al. } \\cite{chaitanya2020contrastive} achieved the next best performance with DSC values of 56.9\\% ($L=10\\%$), 69.4\\% ($\\sim$10\\% less than ours) with 20\\% labeled data and 79.4\\% with 40\\% labeled data.\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale = 0.5]{segoutputs.pdf}\n    \\caption{Visual comparison of our results with SoTA methods and ground truth, thus qualitatively validating the superiority of IDEAL in terms of segmentation performance.}\n    \\label{fig:segoutputs}\n\\end{figure}\n\n\n\n\\section{Conclusion}\nIn this paper, we propose a semi-supervised approach for cardiac MRI segmentation to tackle the labeling bottleneck in medical imaging. Our framework leverages self-supervised contrastive learning with a novel projection head to capture dense local feature representation during pre-training, followed by employing a unique cross-consistency regularization scheme during model fine-tuning for the downstream segmentation task. Results show that our IDEAL framework outperforms several state-of-the-art methods on two widely used cardiac MRI datasets. We believe that such a paradigm can also be employed in cross-modal and cross-domain scenarios, as well as extended to natural images, something we intend to explore in near future.\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\n\nThe extended \\hi\\ disks of spiral galaxies exhibit characteristic 21 cm line\nwidths of 5--15\\ \\kms\\ on small scales.  Line widths are observed to be\n$\\sim6$--10~\\kms\\ even beyond $r_{25}$, where the star formation rate (SFR)\nand the associated feedback are presumably inefficient\n\\citep*{vanderkruit1982, Dickey1990, Kamphuis1993, Rownd1994, vanzee1999}.  In\nsome observed cases, the \\hi\\ disks exhibit an overall outward decrease of the\nvelocity dispersion to 6--8\\ \\kms\\ \\citep{Boulanger1992,Petric2007}.\n\nThe observed line widths are attributable in part to thermal broadening and in\npart to turbulence.  At pressures typical of disk galaxies, interstellar\natomic gas has two temperatures at which it can remain in stable thermal\nequilibrium \\citep{Wolfire1995}. The gas is observed to be distributed with\nroughly 60\\% of the mass evenly distributed between the equilibrium\ntemperatures \\citep{Kulkarni1987, DickeyLockman1990, Dickey1993} and 40\\%\ntransiting between them \\citep{Heiles2001}.  The cold ($\\sim100$~K) neutral\ngas emits lines with a characteristic line width of $\\sim1$~\\kms, while the\nwarm ($\\sim8000$~K) neutral gas has a line width of $\\sim8$~\\kms.  Spectral\nlines observed to be broader than $\\sim8$~\\kms\\ can be attributed to\nturbulence -- stochastic fluctuations of velocity and pressure on scales much\nlarger than the mean free path of the gas.  Heating by a galactic interior or\nextragalactic background of ultraviolet (UV) radiation \\citep{Schaye2004} may\nbe significant in the outermost regions of \\hi\\ disks, where velocity\ndispersions approach thermal values for the warm phase.\n \nThere are several energetic sources that can induce turbulent motions in the\ninterstellar medium (ISM), among which feedback from recently formed stellar\npopulations and magnetorotational instabilities\n\\citep[MRI;][]{Balbus1991,Balbus1998,Sellwood1999} are some of the most\nimportant.  The stirred turbulence drives motions at smaller and smaller\nscales until viscosity dissipates the motion into thermal energy.  This\nhierarchical description was formulated by \\citet{Kolmogorov1941} for an\nincompressible turbulent fluid, though it applies well to the small scale\ndynamics of the gas in galaxy disks \\citep*{Elmegreen2003}.  Ultimately, the\nISM radiates away the thermal energy, e.g, by line cooling.  Since the typical\nradiative cooling time \\citep[$\\sim10^3$~years;][]{Wolfire2003} is shorter\nthan the time scale for turbulent energy to decay\n\\citep[$\\sim10$~Myr;][]{maclow1999}, the turbulent energy must be continuously\nreplenished on the decay time scale.  The direct identification of the energy\nsources that sustain the turbulence at this rate has drawn considerable\ninterest for well over three decades, but has not yet been concluded\n\\citep[e.g.][]{maclow2004, elmegreen2004}.\n\nNumerical simulations have demonstrated that both hydrodynamical and\nmagnetohydrodynamic turbulence decay over a relatively rapid time scale,\ncomparable to the local dynamical time, i.e. $\\sim 10$~Myr \\citep*{maclow1998,\n  Stone1998, maclow1999, Padoan1999, Avila2001, Ostriker2001}, thus requiring\na driving mechanism that continuously provides energy to the medium to\nmaintain a steady state \\citep{Gammie1996, maclow1999}.\n\nWhich mechanisms drive ISM turbulence remains a matter of debate.  As first\nenvisioned by \\citet{Spitzer1978}, star formation itself injects energy back\nto the ISM and drives turbulence through stellar winds of O\/B and Wolf-Rayet\nstars, UV ionizing radiation, and supernova (SN) explosions\n\\citep*[e.g.][]{Chiang1985, Rosen1993, Kim2001, Dib2006, Joung2006,\n  Avillez2007}. The energy input from SNe likely dominates stellar feedback\n\\citep{Kornreich2000, maclow2004}.  The overall galaxy kinematics can also\ngenerate some turbulence through spiral wave shocks \\citep{roberts69},\nrotational shear \\citep{Schaye2004}, swing amplified shear instabilities\n\\citep*{Huber2001,Wada2002}, gravitational potential energy of the arms\n\\citep{Elmegreen2003}, and non-linear development of gravitational instability\n\\citep*{li05,li06}.  Magnetized disk instabilities including the\nmagnetorotational instability \\citep*{Balbus1991, Balbus1998,\n  Sellwood1999,Dziourkevitch2004, Piontek2005}, and the magneto-Jeans\ninstability \\citep{Wada2004, Kim2006, Kimetal2006}.  Finally, there could also\nbe a contribution from thermal instability \\citep{\n  Koyama2002,Kritsuk2002,Dib2005,Hennebelle2007}.\n\n\nIn this paper we attempt a direct observational identification of the energy\nsources that drive the turbulence. Our results suggest that the energy input\nfrom young stars determines the \\hi\\ line width within the star forming region\nof galactic disks, while at large radii, where star formation is absent or\nfeeble, the line width is either sustained by the MRI \\citep{Sellwood1999}, or\nby thermal broadening from UV heating.  We examine whether SN feedback, gas\ntemperature, and MRI can maintain effectively the observed ISM line width,\nassuming that the energy loss occurs on the self-dissipation timescales\npredicted for the turbulence \\citep[][]{maclow1999}.  \\citet{maclow2004}\\ \nargue that SNe, among other mechanisms of energy injection, are the most\neffective in actively star forming regions, while in the outermost regions of\ngalaxy disks, where the SFR is low or zero, the MRI becomes important in\nsustaining both the magnetic field, even in low density regions, and the gas\nvelocity dispersion \\citep{Sellwood1999}.\n\n\nWe use multi-wavelength data in the radio, infrared (IR), and far UV\n(\\S~\\ref{sec:data}), to compare detailed maps of the ISM kinetic energy\ndensity implied by 21~cm line widths to maps of the SFR density for a sample\nof nearby spiral and dwarf galaxies. In this comparison, we take the SFR\ndensity as a proxy for the energy input rate to the ISM.  Relying on the\nmeasurement of the velocity dispersion as a tracer of turbulence in the\nneutral gas, we look at both the correlation between SFR and turbulent motions\nin the \\hi\\ on a pixel-by-pixel basis and at their mean values as function of\ngalactocentric distance (\\S~\\ref{sec:analysis}).  We find that the azimuthal\naverage of the \\hi\\ velocity dispersion decreases radially much more slowly\nthan the SFR (\\S~\\ref{sec:results}); specifically, the radial profile of \\hi\\ \nvelocity dispersion approaches the \\hi\\ thermal value for the warm neutral\nmedium where the radial profile of SFR truncates.  Therefore, we investigate\n(1) pixel-by-pixel how the kinetic energy of the gas turbulence correlates\nwith the SFR, especially in terms of SN rate, and (2) what role thermal\nbroadening and MRI play in regions of low SFR and considerable velocity\ndispersion (\\S~\\ref{sec:discussion}). In particular, we compare the current\nturbulent kinetic energy of the gas turbulence to the energy input from both\nSNe and MRI (and the efficiencies of this input), on the basis that the\nturbulence dissipation timescales, provided by extensive theoretical modeling,\napproximate the energy loss timescales.\n\n\n\n\\section{DATA}\\label{sec:data}\n\nThe main observational data for our analysis are maps of the ISM kinetic\nenergy surface density derived from high resolution \\hi\\ data.  For a\nsub-sample we include carbon monoxide (CO) emission maps to explore the\ncontribution of the molecular gas to the kinetic energy budget. These maps are\nthen compared, pixel by pixel, to analogous maps of the SFR surface density,\nderived from UV and thermal IR maps for the same objects.\n\nWe use maps produced by The \\hi\\ Nearby Galaxy Survey\n\\citep[THINGS;][]{walter2008}, a survey of the 21 cm emission line carried out\nwith the NRAO\\footnote{The National Radio Astronomy Observatory is a facility\n  of the National Science Foundation operated under cooperative agreement by\n  Associated Universities, Inc.}\\ Very Large Array, to obtain the column\ndensity \\shi\\ and velocity dispersion \\sighi\\ of neutral hydrogen.  The THINGS\ndata have an angular resolution of $\\sim7''$, and a velocity resolution of\neither 2.6 or 5.2~\\kms\\ (see Table~\\ref{tab:objs}), and a $3\\,\\sigma$\nsensitivity corresponding to a column density of $N({\\rm HI})\\ge 3.2\n\\times10^{20}$~cm$^{-2}$. To avoid projection effects in the velocity\ndispersion maps, we consider only those galaxies that are more face-on than\n$\\sim 50$\\dg\\ \\citep[see, e.g.,][]{Leroy2008}, leaving 11 disk galaxies in\ntotal, including three dwarf galaxies (Holmberg~II, IC~2574, and NGC~4214),\nlisted in Table~\\ref{tab:objs}.\n\nWe complement the analysis of the \\hi\\ gas with the mass surface density and\nvelocity dispersion of molecular hydrogen (H$_2$) gas for about half of our\ngalaxies: NGC~628, NGC~3351, NGC~3184, NGC~4214, NGC~4736, and NGC~6946.  The\nH$_2$ maps are derived from CO $J=2\\rightarrow1$ emission maps, obtained at\nthe IRAM 30-m telescope by \\citet{Leroy2008b}\\ using the HERA focal plane\narray. These maps have resolution of $\\sim11''$ and 2.6~\\kms, and sensitivity\nof $\\Sigma_{\\rm H2}\\ge 4$~\\msun~pc$^{-2}$. To derive $\\Sigma_{\\rm H2}$ we\nassume a conversion factor $X_{\\rm CO}=2\\times10^{20}$~cm$^{-2}$~K~\\kms\\ and a\nline ratio $N(J=2\\rightarrow1)\/N(J=1\\rightarrow0)=0.8$.\n\nThe SFR density maps were derived by \\citet{Leroy2008} from 24~\\mum\\ and\nfar-UV (FUV) maps, combining dust-enshrouded star formation activity and\nunobscured star formation, respectively.  The MIPS 24~\\mum\\ emission maps\n\\citep{Rieke2004}\\ are taken from the Spitzer Infrared Nearby Galaxies Survey\n\\citep[SINGS;][]{kennicutt03}, and the FUV emission maps are taken from the\nGalaxy Evolution Explorer Nearby Galaxy Survey\n\\citep[GALEX-NGS;][]{gildepaz2007}, for a set of disk galaxies in common with\nthe THINGS sample.  The MIPS instrument has a resolution of $\\simeq6''$ at\n24~\\mum, and a wide areal coverage, which typically covers $\\sim2\\,r_{25}$.\nThe GALEX FUV images have a resolution (FWHM) of $5.6''$ within the\n$\\lambda=1350-1750$~\\AA\\ band and have a field of view of $\\sim1.25$\\dg.\n\n\\section{DATA ANALYSIS}\\label{sec:analysis}\n\nAs a first step in comparing the kinetic energy of the gas to possible drivers\nof turbulence, we convert the previously described maps (\\S~\\ref{sec:data}) of\n\\hi\\ and H$_2$ gas column density and kinematics, and the 24~\\mum~+~FUV\nsurface brightness into physical quantities. Specifically, we obtain mass\nsurface density, \\shi, $\\Sigma_{\\rm H2}$, velocity dispersion, \\sighi,\n$\\sigma_{\\rm H2}$, and SFR surface density, \\sfr.  Afterwards, we proceed with\na comparison between the turbulence kinetic energy, obtained from the gas mass\nsurface density and velocity dispersion, and the SFR on a pixel-by-pixel\nbasis.\n\n\\subsection{Mass Surface Density and Velocity Dispersion}\\label{sec:moments}\n\nWe determine the \\hi\\ mass surface density, \\shi, and the velocity dispersion,\n\\sighi, by taking the moments of fully reduced data cubes for our 11 sample\ngalaxies \\citep[the prior reduction steps of are described in][]{walter2008}.\nWe obtain the pixel-by-pixel maps of $\\Sigma_{\\rm HI}(x,y)$, $\\bar{v}(x,y)$,\nand $\\sigma(x,y)$, by calculating integrals over the data cubes for all the\npositions on the sky $S_i(x,y,v_i)$.  All the sums are calculated over those\nchannels with signal identified by the THINGS masking \\citep{walter2008}; we\nonly calculated moments for those ($x,y$) line-of-sight positions with signal\nin at least 3 channels.\n\nThe zero-th moment, $\\Sigma_{\\rm HI}$, is calculated by integrating the \\hi\\ \ndata cube along the velocity dimension for a total number $N$ of velocity\nchannels using:\n\\begin{equation}  \\label{eq:m_zero}\n  \\Sigma_{\\rm HI} =  \\sum_{i=1}^N S_i,\n\\end{equation}\nwhere $S_i$ denotes the signal within the $i$-th velocity channel. Afterwards,\n\\shi\\ is corrected for inclination; the adopted inclinations are listed in\nTable~\\ref{tab:objs}.  For each sample galaxy, the spectral dimension of the\n\\hi\\ data cubes is sufficiently broad to encompass all relevant velocity\nvalues.\n\nThe first moment map, the intensity-weighted mean velocity along the line of\nsight, is given by:\n \\begin{equation}\\label{eq:m_one}\n\\bar{v} =\\frac{1 }{\\Sigma_{\\rm HI}}\\;\\sum_{i=1}^N v_i\\,S_i,\n\\end{equation}\nwhere $v_i$ is the velocity value of the $i$-th channel. We only use $\\bar{v}$\nin this context to calculate the central 2nd-moment.\n\nThe variance, corresponding to the square of the line-of-sight velocity\ndispersion, is given by the intensity-weighted mean deviation:\n \\begin{equation}\\label{eq:mom_two}\n\\sigma^2\n=\\frac{1 }{\\Sigma_{\\rm HI}}\\;\\sum_{i=1}^N  (v_i-\\bar{v})^2\\,S_i.\n\\end{equation}\n\nThroughout our analysis we restrict ourselves to the moment definition of\nEq.~\\ref{eq:mom_two}\\ to define the variance.  We presume that $\\sigma$\n(Eq.~\\ref{eq:mom_two}) is the most sensible tracer of turbulent or more\nprecisely, disordered motions, since it is applicable to any line profile.  We\naddress the discussion of this assumption in \\S~\\ref{sec:realities}.\n\n\n\nWe complement the \\hi\\ maps with H$_2$ column density and kinematics data\nobtained from CO emission (\\S~\\ref{sec:data}). Following the same procedure\npreviously adopted to calculate the relevant \\hi\\ maps, we obtain the mass\nsurface density, $\\Sigma_{\\rm H2}$, and the velocity dispersion, $\\sigma_{\\rm\n  H2}$, for the H$_2$ gas.  As described in the following sections, while\nH$_2$ is an important contributor to the total mass surface density, its\ncontribution to the total turbulent kinetic energy of gas turns out to be\nrather small in our sample.\n\n\\subsection{Realities of Measuring the Velocity \n  Dispersion in THINGS}\\label{sec:realities}\n\nIn the case of perfect signal, the second moment is exactly the kinetic\nenergy of \\hi\\ gas. Even at limited signal-to-noise ratio ($\\rm S\/N$) but with\nnormally-distributed noise, we expect the measured variance to scatter about\nthe true value in a well-behaved way, so that by averaging measurements over\nenough area, we can approximate the true value.\n\n\nAs an alternative approach, assuming and then fitting a fixed line profile\nshape (e.g., a Gaussian) fails in two ways. First, towards the centers of\ngalaxies, line profiles tend to be broad and often contain multiple emission\ncomponents. Second, in the outskirts of galaxies, where individual lines of\nsight tend to have relatively low $\\rm S\/N$, a simple Gaussian fit tends to\nunderestimate the line width by latching onto only the peak of the emission.\nIn the low $\\rm S\/N$ regime, the line profiles are approximately Gaussian.\nFor a few sample galaxies (NGC~3184, NGC~5194, NGC~6946, and NGC~7793) we find\nthat the Gaussian $\\sigma$ value is systematically lower than the second\nmoment. For these galaxies, however, the difference between Gaussian $\\sigma$\nand 2nd-moment is lower than $\\sim 2-3$~\\kms\\ in average. Since we do not know\na priori the \\hi\\ gas line-of-sight velocity distribution, and the profiles\nare not necessarily Gaussian, the best tracer of kinetic energy, as a model\nindependent estimate, is the variance.\n\nTherefore, we avoid fitting and attempt to minimize systematics by measuring\nthe second moment only on regions that have been identified to contain signal.\nThe production of these ``blanked cubes'' is described in \\citet{walter2008},\nand is briefly summarized here. First, the cubes were smoothed to 30'' angular\nresolution. Then, all areas without $\\rm S\/N> 2$ in 2 consecutive velocity\nchannels are blanked (set to 0). This does an excellent job of identifying all\nareas with significant emission but still left some false positives. These\nwere removed by a by-eye inspection that focused on agreement with the overall\nvelocity structure and morphology of the galaxy. We test the systematics\ninduced by such blanking on a face-on galaxy by measuring $\\sigma$ on the\nblanked cube and then expanding the blanking mask and remeasuring $\\sigma$. We\nfind that small expansions of the mask on both directions along the velocity\ndimension, e.g. by $\\pm 15$~km~s$^{-1}$, have almost no impact on the measured\ndispersion. Larger expansion begin to show systematic effects that we want to\navoid (particularly at low $\\rm S\/N$).\n\nThese blanked cubes minimize the effect of artifacts in the THINGS cubes and\nyield a more robust estimate of kinetic energy than fitting an assumed line\nprofile. Therefore, the profiles of $\\sigma$ and $E_k$ represent the best\npossible estimates using the available data. However, we emphasize that a\nrigorous attempt to match other observations or simulations to our\nmeasurements should bear in mind (and ideally duplicate or simulate) this\nblanking procedure.\n\n\\subsection{Turbulent Kinetic Energy}\n\nWith the moment definition of \\shi\\ and \\sighi, the \\hi\\ kinetic energy per\nunit area is given by $E_k=3\/2\\;\\Sigma_{\\rm HI}\\;\\sigma_{\\rm HI}^2$, where the\nfactor 3\/2 takes into account all three velocity components assuming that the\nvelocity dispersion is isotropic.  For the galaxies with molecular gas maps,\nwe define the total kinetic energy as $E_k=3\/2\\times(\\Sigma_{\\rm\n  HI}\\;\\sigma_{\\rm HI}^2+\\Sigma_{\\rm H2}\\;\\sigma_{\\rm H2}^2)$.  In\n\\S~\\ref{sec:results}\\ we describe the relationship between the \\hi\\ (or\n\\hi~+~H$_2$) gas kinetic energy $E_k$ and the \\sfr\\ maps in a pixel-by-pixel\nscatter plot.\n\n\\subsection{Star Formation Rate}\\label{sec:limitations}\n\nWe use maps of SFR surface density, \\sfr, derived by \\citet{Leroy2008}, from\ncombining the 24~\\mum\\ and the FUV emission maps taken from the SINGS and the\nGALEX-NGS surveys (\\S~\\ref{sec:data}).  \\citet{Leroy2008}\\ have calibrated the\nSFR represented by the UV and IR emission by comparing these maps to \\halpha,\n24~\\mum, and Pa$\\alpha$ emission from \\hii\\ regions\\ and young compact stellar\nclusters \\citep{calzetti07}.  The 24~\\mum\\ band emission is mostly radiation\nfrom hot dust heated by the UV light from young massive stars and therefore\ntraces dust-enshrouded, ongoing star formation over a time scale $3-10$~Myr\n\\citep{Calzetti2005,perez2006,tamburro2008}, although part of the emission\nproceeds from outside the \\hii\\ regions, thus tracing older ($>10$~Myr)\nstellar populations.  FUV emission is mostly photospheric emission from O and\nB stars. It thus complements the 24~\\mum\\ emission in regions poor in dust\ncontent, probing therefore low-metallicity and older regions of star formation\nover timescales of $\\tau\\sim10-100$~Myr \\citep{Calzetti2005,Salim2007}.\n\n\n\nThe calibration provides SFR estimates with an uncertainty of 40\\% at most at\nhigh SFR, depending on variations in geometry, dust temperature, and age of\nstellar populations \\citep{Leroy2008}.  At low SFR, a substantial part of the\n24~\\mum\\ emission proceeds from diffuse dust in the ISM, which may not be\nassociated directly to recent star formation, with the dust being heated by\nnearby (young and old) star clusters.  \\citet{Leroy2008} argue that below a\nfiducial threshold of $\\Sigma_{\\rm SFR}=10^{-10}\\; \\rm\nM_\\odot\\;yr^{-1}\\;pc^{-2}$ the SFR maps represent upper limits to the true SFR\nbecause of the contribution of this diffuse dust component.\n\n\n\n\n\\section{RESULTS}\\label{sec:results}\n\nIn this section we describe our two main results: (1) the observation that a\nradial decline of the \\hi\\ velocity dispersion is pervasive throughout the\nentire sample with little dependence on galaxy type; and (2) that we find\ncorrelation between the kinetic energy of gas and SFR with a slope close to\nunity and a similar proportionality constant in all objects.  Both results\ndepend on the exceptional quality of the data in terms of spatial and velocity\nresolution and the wide field of view. We reserve the broader interpretation\nof our results for \\S~\\ref{sec:discussion}.\n\nIn the following, we use units of $r_{25}$, which provides a convenient\nnormalization. \\citet{Leroy2008, tamburro2008}\\ measured the exponential scale\nlengths of near-IR emission from our sample, a good proxy for stellar mass.\n$r_{25}$ is typically $4.6\\pm0.8$ times the near-IR scale length, and the SFR\nmaps yield comparable scale lengths to the near-IR. When we normalize by\n$r_{25}$ then, it is roughly equivalent to normalizing by the scale length of\nthe disk.\n\n\\subsection{Radial Profiles of $\\sigma_{\\rm HI}$}\\label{sec:radial}\n\nTo start, we determine the azimuthally averaged radial profiles of \\hi\\ \nvelocity dispersion, SFR, and gas kinetic energy, by calculating the average\nvalues of \\sighi\\ and SFR within annuli of $\\sim15''$ width.  The resulting\n\\sighi$(r)$ and \\sfr$(r)$ profiles are shown in Fig.~\\ref{fig:m2_vs_r_fit}.\nThey exhibit a radial decline of \\hi\\ velocity dispersion as a common\ncharacteristic for all sample galaxies independent of their dynamical mass.\nWhile previous studies have reported this individually for a few disk galaxies\n\\citep[i.e.,][]{Boulanger1992, Petric2007}, we show for the first time that\nthis is true for a significant sample of dwarf and normal spiral galaxies.\n\nTo characterize the \\sighi\\ radial gradients, we fit a linear relation to the\nradial profiles of \\sighi\\ for all $r\\ge r_{25}$, a regime where $\\sigma(r)$\nis approximately linear with $r$ for all the galaxies of the sample. With\n$\\sigma(r)$ going from $\\gtrsim20$~\\kms\\ down to $\\sim5\\pm2$~\\kms\\ near the\noutermost observed radius (Fig.~\\ref{fig:m2_vs_r_fit}), the velocity\ndispersion decreases with radius by $\\simeq3$--5~\\kms\\ per $\\Delta r_{25}$;\nfor comparison, azimuthal variations at a fixed radius are $\\simeq 5$~\\kms.\nIn Fig.~\\ref{fig:s25_avg}, we display and summarize the intercept value of\n\\sighi\\ at $r_{25}$ and the \\hi\\ mass weighted median of \\sighi\\ for all\nsample galaxies individually. Our observations resolve the outward radial\n\\sighi\\ decline well, since a typical $r_{25}$ for our sample galaxies\ncorresponds to $\\sim30$--40 times the 7'' \\hi\\ resolution limit and the\ntypical radial extent of the \\hi\\ emission is $2-4\\times r_{25}$.\n\nRemarkably, we find that all galaxies have the same \\hi\\ velocity dispersion\nat their respective $r_{25}$: a general value of\n$\\sigma(r_{25})\\simeq10\\pm2$~\\kms, which displays no apparent trend with the\ndynamical mass and morphological type, and is consistent with the \\hi\\ \nmass-weighted median value $\\langle\\sigma\\rangle$ (see Fig.~\\ref{fig:s25_avg}\\ \nand Table~\\ref{tab:objs}).  The radial \\sighi\\ gradient has in all cases the\nsame sign as the much steeper decline of the mean SFR as function of radius as\nshown for comparison in logarithmic scale in Fig.~\\ref{fig:m2_vs_r_fit}.\n\nSince the sample galaxies are more face-on than 50\\dg, the systematic increase\nof \\sighi\\ towards the center cannot be due to increasing beam smearing\neffects.  We analyze the combined effect of beam smearing, inclination and\nrotation curve on the velocity dispersion by constructing a sample data cube\ncontaining signal in only one velocity channel per each ($x,y$) spatial\nposition (i.e. intrinsic \\sighi~=~0) and characterized by the same inclination\nand rotation curve of NGC~5055 -- the most inclined disk of the sample with a\nfast rotation speed.  After convolving this sample data cube with a kernel of\n7'' FWHM (the resolution of our \\hi\\ maps), we calculate the velocity\ndispersion using second moments (Eq.~\\ref{eq:mom_two}) as done throughout our\nanalysis (\\S~\\ref{sec:moments}).  The resulting line broadening from beam\nsmearing for a disk with the rotation curve and orientation of NGC~5055 is\nonly $\\lesssim 5 $~\\kms, lower than the observed velocity dispersion in\nNGC~5055 by $\\sim10$--20~\\kms.  The fractional contribution of the velocity\ndispersion from beam smearing to the total observed velocity dispersion is\nonly 20\\% in the central region ($r<1\/4\\; r_{25}$) and at most 10\\% at larger\ngalactocentric radii.\n\n\n\\subsection{The H{\\sc i} Kinetic Energy Density as a Function of Radius}\\label{sec:radialek}\n\nThe \\hi\\ kinetic energy density, $E_k$, in each pixel exhibits a clear radial\ndecline, as shown in the full pixel-by-pixel distribution\n(Fig.~\\ref{fig:Ek_vs_r}), where the black contours and color scale indicate\nthe density of pixels at each \\hi\\ kinetic energy and radius, while the red\ncontours show the sum of atomic and molecular kinetic energy.  The latter\ndiverges from the \\hi\\ kinetic energy in the inner parts of some galaxies,\nindicating that the cold molecular gas contributes to the kinetic energy\nbudget. Since every galaxy of the sample includes $\\sim2\\times10^5$ pixels\nwith significant signal, we display in Fig.~\\ref{fig:Ek_vs_r} the density\ncontours of the data points.\n\nWe include the analysis of the H$_2$ mass surface density and velocity\ndispersion derived from the CO emission for a few sample galaxies\n(\\S~\\ref{sec:analysis}), to quantify the contribution of the molecular gas to\nthe total kinetic energy at high H$_2$-to-\\hi\\ mass ratio.  The total kinetic\nenergy $E_k=E_{\\rm HI}+E_{\\rm H2}$ is plotted in Fig.~\\ref{fig:Ek_vs_r}\\ (with\nred contours) for the galaxies with CO data. For the galaxies NGC~4736,\nNGC~5055, and NGC~6946, $E_{\\rm H2}$ is comparable to $E_{\\rm HI}$ or even\ndominant in the central regions of galaxy disks; for these galaxies the H$_2$\nmass and spatial extent is considerable. For those galaxies where the \\hi\\ gas\ndominates the total gas mass, i.e., NGC~628, NGC~3184, NGC~3351, and the dwarf\ngalaxies Holmberg~II, IC~2474, and NGC~4214, characterized by little or no\ndetected molecular gas, the molecular gas does not contribute much to the\ntotal kinetic energy.\n\n\n\n\\subsection{Correlation between $E_k$ and \\sfr}\\label{sec:pixel}\n\nIn Fig.~\\ref{fig:Ek_vs_sfr} we compare pixel-by-pixel the relation between the\nkinetic energy density of the \\hi\\ gas, $E_k=3\/2\\,\\Sigma_{\\rm HI}\\,\\sigma_{\\rm\n  HI}^2$, and the SFR surface density -- a proxy for the energy input rate by\nSN.  We find that in all galaxies these quantities are well correlated with a\nslope close to unity and with no evident dependence on dynamical mass\n(Fig.~\\ref{fig:Ek_vs_sfr}).  Note that a considerable fraction of all data\npoints in Fig.~\\ref{fig:Ek_vs_sfr}\\ lie below the noise estimated for the\n\\sfr\\ maps (\\S~\\ref{sec:limitations}). These data points lie in the outermost\nparts of galaxy disks ($r>2\\times r_{25}$), where there is little or no\nongoing star formation (cf. Fig.~\\ref{fig:Ek_vs_r}).\n\nWe note that the slope of all correlations in Fig.~\\ref{fig:Ek_vs_sfr}\\ \nflattens at high $E_k$ and \\sfr, although we argue in \\S~\\ref{sec:radialek}\nthat this is not caused by a higher abundance of molecular gas at higher \\sfr.\nThe total kinetic energy $E_k=E_{\\rm HI}+E_{\\rm H2}$ is plotted in\nFig.~\\ref{fig:Ek_vs_sfr} (with red contours) for the six sample galaxies with\nH$_2$ data. Only for NGC~5055 and NGC~6946 is the kinetic energy of the\nmolecular gas, $E_{\\rm H2}$ important; in those cases the $E_{\\rm HI}+E_{\\rm\n  H2}$ vs.\\ \\sfr\\ relation is linear and the slope is close to unity.\n\n\\section{DISCUSSION}\\label{sec:discussion}\n\nWhat scenario does our data support for producing the observed line widths?\nSN-driven turbulence seems likely to be the dominant factor broadening\nline widths within the radius of active star formation, since we find that the\nlevel of predicted SN energy is sufficient to account for the turbulent\nkinetic energy implied by the line width as a function of radius.\n\nThe radial slopes of SN energy and kinetic energy of the neutral (\\hi\\ and\nH$_2$) gas agree qualitatively so that the kinetic energy of the gas is\nproportional to the local star formation rate.  Yet, the fact that the \\hi\\ \nvelocity dispersion approaches its thermal value of roughly 6~\\kms\\ well\nbeyond the radius of detectable star formation indicates that either (1) the\nline broadening is due to UV heating, with a warm neutral medium temperature\nof $\\sim 5000$~K resulting in \\sighi~$\\sim 6$~\\kms, or (2) the gas is actually\nturbulent and another mechanism such as the MRI is driving the turbulence.  We\nnow explore whether our new data support this scenario.\n\nIn the following, we compare the observed line widths with the most plausible\nmechanisms for generating them: SNe (\\S~\\ref{sec:supernova}), UV heating\n(\\S~\\ref{sec:thermal}), and MRI (\\S~\\ref{sec:mri}). SNe and MRI both produce\nbroad line widths by driving turbulence, while UV heating can produce thermal\nbroadening.  The required energy injection rate depends both on the kinetic or\nthermal energy of the gas, which can be derived from the observed line widths\nand the turbulence decay or cooling timescales, which must be derived from\nmodels.  More precisely, the gas kinetic energy implied by the linewidth\nconsists of a combination of turbulence and the thermal energy associated with\nthe (warm) gas temperature, i.e. $E_k=E_{\\rm turb}+E_{\\rm therm}$.  If the\nthermal broadening is much less effective than the turbulence, then $E_k\\simeq\nE_{\\rm turb}$.\n\nIf the gas turbulence is mainly driven by SNe and MRI, then we expect the\ndissipation rate of turbulence to equal the sum of the energy input rates of\nSNe and MRI,\n\\begin{equation} \n\\dot{E}_k \\simeq \\epsilon_{\\rm SN}\\,E_{\\rm\n  SN} \/\\tau_{\\rm SN}+ \\epsilon_{\\rm MRI} E_{\\rm MRI} \/ \\tau_{\\rm MRI},\n\\end{equation}\nwhere $\\epsilon_{\\rm SN}$ and $\\epsilon_{\\rm MRI}$ are the efficiencies, and\n$\\tau_{\\rm SN}$ and $\\tau_{\\rm MRI}$ are the decay times of turbulence driven\nby the two mechanisms.  Different mechanisms can result in different decay\nrates because they have different driving scales and magnitudes\n\\citep{Stone1998,maclow1999}.\n\n\n\\subsection{Supernova Energy}\\label{sec:supernova}\n\nAssuming steady state equilibrium between the energy input rate from SNe to\nturbulent gas motions and the energy loss rate from dissipation of this\nturbulence, then the resulting kinetic energy $E_k = \\epsilon_{\\rm SN}\n\\dot{E}_{\\rm SN} \\tau_{SN}$, where $\\dot{E}_{\\rm SN}$ is the rate of released\nSN energy, which we estimate from the SFR, and the SN feedback efficiency\n$\\epsilon_{\\rm SN}$ is the fraction of SN energy converted to turbulent\nmotions in the cold gas.  \\citet{maclow1999} finds that the dissipation rate\nof turbulence depends on the driving scale $\\lambda$ and the velocity\ndispersion $\\sigma$ as\n\\begin{equation}\n\\tau_D\\simeq9.8\\;(\\lambda_{100}\/\\sigma_{10})\\;\\rm Myr,\n\\end{equation}\nwhere $\\lambda_{100} = \\lambda \/ 100$~pc and $\\sigma_{10} = \\sigma \/\n10$~km~s$^{-1}$.  Numerical simulations of SN-driven turbulence yield\n$\\lambda=100\\pm30$~pc \\citep{Joung2006, Avillez2007}, and our own analysis\ngives an average velocity dispersion $\\sigma=10$~\\kms (\\S~\\ref{sec:radial}).\nThe SN energy input rate, $\\dot{E}_{\\rm SN}$, can be estimated from the SN\nrate implied by our SFR maps.  The SN rate per unit area, $\\eta$, depends on\nthe fraction $f_{*\\rightarrow \\rm SN}$ of all recently formed stars that\nterminate in core-collapse SNe:\n\\begin{equation}\n  \\label{eq:sn_rate}\n\\eta= \\frac {\\rm SFR}{\\avg{m}}  \\times  f_{*\\rightarrow \\rm SN},\n\\end{equation}\nwhere $\\avg{m}$ is the average mass of stars of the population.  We assume\nthat only those stars in the mass range ($8-120$)\\,\\msun\\ can form\ncore-collapse SNe. The SFR maps used in our analysis \n  assume an initial mass function (IMF)\n$\\phi(m)=m^{-\\alpha}$, where $\\alpha=1.3$ for the mass range\n($0.1-0.5$)\\,\\msun\\ and $\\alpha=2.3$ for the mass range ($0.5-120$)\\,\\msun\\ \n\\citep[cf.][]{Leitherer1999, calzetti07}.  Then, the SN fraction\n\\begin{equation}\n  \\label{eq:frac_sn}\n  f_{*\\rightarrow \\rm SN}\/\\avg{m}=\n\\frac{\\int_{8\\,\\rm M_\\odot}^{120\\,\\rm M_\\odot}\n  \\phi(m)\\, dm} {\\int_{0.1\\,\\rm M_\\odot}^{120\\,\\rm M_\\odot}\n  \\phi(m)\\, dm} \/ \\frac{\\int_{0.1\\,\\rm M_\\odot}^{120\\,\\rm M_\\odot}\n  m \\phi(m)\\, dm} {\\int_{0.1\\,\\rm M_\\odot}^{120\\,\\rm M_\\odot}\n  \\phi(m)\\, dm},\n\\end{equation}\nyielding $f_{*\\rightarrow \\rm SN}\/\\avg{m}\\simeq1.3\\times10^{-2}\\;\\rm\nM_\\odot^{-1}$.  If we were to reduce the upper mass limit to 50~\\msun\\ yields\n$f_{*\\rightarrow \\rm SN}\/\\avg{m}\\simeq1.2\\times10^{-2}\\;\\rm M_\\odot^{-1}$, and\nan upper mass limit of 20~\\msun\\ yields $0.9\\times10^{-2}\\;\\rm M_\\odot^{-1}$.\nThe effect of these variations on $\\eta$ is unclear, because the upper mass\nlimit of the IMF also affects our translation of UV and IR light into SFR.\nThe UV and IR maps are primarily sensitive to high mass stars. Therefore for a\nlower upper mass limit, therefore, less UV and IR light is emitted per unit\nstar formed and if the upper mass limit is actually lower than we have\nassumed, then we have underestimated the true SFR. In calculating $\\eta$,\nthese higher SFR and lower $f_{*\\rightarrow \\rm SN}\/\\avg{m}$ have opposite\neffects, leaving the impact of changing the upper mass limit on $\\eta$\nunclear. As the upper mass limit decreases, we estimate less intrinsic UV and\nIR emission per high-mass star, which we have used to construct the maps. We\nneglect the contribution of type Ia SNe whose rate is $\\sim1\/3$ of the\ncore-collapse rate for the morphological types of our sample galaxies\n\\citep{Mannucci2005}.  In these circumstances, the SN rate can be\nstraightforwardly calculated as a function of SFR, $\\eta=\\eta(\\Sigma_{\\rm\n  SFR})$, and, depending on the adopted assumptions, the SN rate is\ncharacterized by an uncertainty of a factor $\\sim1\/3$.  We assume that for\neach SN explosion only a fraction $\\epsilon_{\\rm SN}\\le1$ of $10^{51}$~erg --\nroughly the energy released by a single SN event \\citep{Heiles1987} -- is\nconverted into turbulence. Then in steady state the kinetic energy of the gas\nturbulence $E_k=\\eta\\times(\\epsilon_{\\rm SN}\\,10^{51}\\;{\\rm erg})\\tau_D$.\n\nIn Fig.s~\\ref{fig:Ek_vs_r} and~\\ref{fig:Ek_vs_sfr}, we compare our estimate\nfor the total energy from SNe produced within a single turbulent decay time\n$\\tau_D=9.8$~Myr with $E_k$, the kinetic energy derived from the line width,\nas a function of radius and of SFR, respectively, for different values of SN\nefficiency.  Fig.~\\ref{fig:Ek_vs_r} shows the azimuthally averaged SN energy\ndecreasing as a function of radius.  A universal $\\epsilon_{\\rm SN}$ would\nproduce a linear correlation between $E_k$ and \\sfr\\ as shown in\nFig.~\\ref{fig:Ek_vs_sfr}, where lines of unity slope and constant values of\n$\\epsilon_{\\rm SN}$ represent SN energy input as a function of SFR.\nFig.~\\ref{fig:Ek_vs_r} and Fig.~\\ref{fig:Ek_vs_sfr}\\ show that, at least in\nthe high star formation regime ($\\Sigma_{\\rm SFR} >10^{-9}\\; {\\rm\n  M}_{\\odot}$~yr$^{-1}$~pc$^{-2}$), the data points typically agree with an\nefficiency $0.1\\le\\epsilon_{\\rm SN}\\times(10^7\\;{\\rm yr}\/\\tau_D)\\le1$.\n\nThis estimate of the efficiency is consistent with numerical simulations that\nestimate $\\avg{\\epsilon_{\\rm SN}}\\simeq0.1$ \\citep{Thornton1998}.  In the low\nSFR regime ($\\Sigma_{\\rm SFR} < 10^{-9}\\; {\\rm\n  M}_{\\odot}$~yr$^{-1}$~pc$^{-2}$), many data points in\nFig.s~\\ref{fig:Ek_vs_r} and~\\ref{fig:Ek_vs_sfr} lie close to the line of\nmaximum efficiency $\\epsilon_{\\rm SN}=1$.  At $\\Sigma_{\\rm SFR} <\n10^{-10}\\;{\\rm M}_{\\odot}$~yr$^{-1}$~pc$^{-2}$, in particular at large\ngalactocentric radii, the estimated \\sfr\\ rates fall to within the noise (see\n\\S~\\ref{sec:limitations}), whilst the measured $E_k$ is typically well\nconstrained.  Nevertheless, $\\epsilon_{\\rm SN} = 1$ remains problematic, since\nit would imply that all the kinetic energy of SN remnants would be deposited\nas kinetic energy of the gas.  In fact \\citet{TenorioTagle1991}\\ argue that\nthe expected SN efficiency $\\epsilon_{\\rm SN}$ should be at most $\\sim0.5$.\nValues graeter than 0.5, therefore, either imply that other sources inject\nenergy into the ISM (see following sections), or that the dissipation\ntimescales are shorter than we assume. In the next sections, we examine\nalternative mechanisms to explain the line width in these regions.\n\n\n\\subsection{Thermal Broadening}\\label{sec:thermal}\n\nThe temperature of the warm phase of the \\hi\\ is maintained by UV heating.\nHowever, neutral gas never reaches temperatures high enough to explain line\nwidths as high as 10~\\kms, which are instead attributed to supersonic\nturbulent motions \\citep{Wolfire2003, Lacour2005}.  As\n\\sighi~$\\gtrsim10$~\\kms\\ for $r\\lesssim r_{25}$ for all our sample galaxies,\nwe argue that thermal effects can be neglected within the star forming radius.\nIn regions of active star formation, stellar winds and ionizing radiation from\nstars appear less effective together than the SNe from the same stellar\npopulation at driving turbulence \\citep{maclow2004}. Thus, SN driven\nturbulence looks likely to dominate there.\n\nIn the outer regions of \\hi\\ disks, on the other hand, the velocity dispersion\napproaches its thermal value.  There, the warm neutral atomic phase\ntemperature is typically $\\sim5000$~K, as measured, e.g., in the solar\nneighborhood \\citep{Heiles2003, Redfield2004}, where the SFR is a few\n$\\times10^{-9}\\; \\rm M_\\odot\\;yr^{-1}\\;pc^{-2}$.  This temperature gives a\nthermal width of \\sighi~$\\sim6$~\\kms. Typical temperatures of the warm medium\ncan even be as high as $\\sim8500$~K \\citep[corresponding to\n$\\sim8$~\\kms;][]{Wolfire1995}.  If thermal broadening is effective, the\nobserved \\sighi~$\\sim6$~\\kms\\ outside the star forming radius does not\nnecessarily involve turbulence.  However, such temperature levels, especially\nat $r\\gtrsim2\\times r_{25}$, where the SFR is below $10^{-10}\\; \\rm\nM_\\odot\\;yr^{-1}\\;pc^{-2}$ (cf.  Fig.~\\ref{fig:m2_vs_r_fit}), still require a\ncontinuous UV background source warming up the \\hi\\ disks of galaxies. At\n$r>r_{25}$, the source of such UV radiation is presumably not local; it could\nbe extragalactic.\n\n\nThe local thermal pressure estimate and the actual velocity dispersion in the\nouter parts of \\hi\\ disks are consistent with the existence of a warm phase\nfor the gas.  For example, \\citep{Leroy2008} estimate a local pressure\n$P\/k\\sim300$~K~cm$^{-3}$ at $r\\simeq2\\times r_{25}$ for the galaxy NGC~4214,\nassuming hydrostatic equilibrium \\citep[cf.][]{Elmegreen1989}.  At a radius\n$2\\times r_{25}$, the \\hi\\ velocity dispersion in NGC~4214 is $\\sim7$~\\kms,\ncorresponding to a temperature of $\\sim5900$~K, which, solving $P\\propto\nn\\,\\sigma^2$, yields a density $n\\simeq0.05$~cm$^{-3}$. This is consistent\nwith the gas in a warm phase according to Figure~7 in \\citet{Wolfire2003},\nwhere a pressure of $\\sim300$~K~cm$^{-3}$ corresponds to\n$n\\sim0.04$~cm$^{-3}$, at least for the outer parts ($r\\sim 15-18$~kpc) of the\nMilky Way, which likely have a similar UV background.  For pressure values\nlower than $300$~K~cm$^{-3}$, all the gas is warm.\n\n\n\\subsection{MRI Energy}\\label{sec:mri}\n\nIf external UV heating is insufficient to maintain a warm phase, then a\nnon-stellar energy source is needed to explain the observed turbulence\n(\\S~\\ref{sec:radialek}). MRI is a plausible candidate. It develops in\ndifferentially rotating disks with angular velocity decreasing outwards, as in\nall but the smallest galactic disks, as long as some weak magnetic field\nthreads the disk.  It has been argued to sustain both the interstellar gas\nturbulence and the galactic magnetic field to a few $\\mu$G \\citep{Piontek2005,\n  Piontek2007}. MRI requires a minimum magnetic field as low as $10^{-25}$~G\nto originate, which is much lower than the seed galactic magnetic fields\n\\citep{Kitchatinov2004}.\n\nFollowing the same line of reasoning as \\S~\\ref{sec:supernova}, if we assume\nthat a fraction $\\epsilon_{\\rm MRI}\\le1$ of the MRI energy is transformed into\nkinetic energy, then in steady state the observed kinetic energy must equal\nthe MRI energy input within the turbulence decay time: $E_k=\\epsilon_{\\rm\n  MRI}\\,\\dot{E}_{\\rm MRI}\\;\n\\tau_{\\rm MRI}$. Theoretical calculations \\citep{Sellwood1999, maclow2004,\n  maclow2008} estimate the production energy rate of MRI to be\n\\begin{equation}\n\\begin{array}{cl}\n\\dot{E}_{\\rm MRI}= &\n3.7\\times10^{-8}\\;{\\rm erg\\;cm^{-2}\\;s^{-1}}\\times \\\\\n & \\;\\left(\\frac{h_z}{100\\;{\\rm pc}}\\right)\\;\n \\left(\\frac{B}{6\\;\\mu{\\rm G}}\\right)^2\\;\\frac{\\Omega}{(220\\;{\\rm Myr})^{-1}},\\\\\n\\end{array}\n\\end{equation}\nwhere $\\Omega\\equiv v\/r$ is the angular velocity, $h_z$ is the vertical\nthickness of the \\hi\\ disk, and $B$ is the magnetic field.  In our analysis,\nwe assume a constant thickness $h_z=100$~pc for all galaxies. This is\nappropriate for the inner Milky Way out to the solar circle\n\\citep{Wolfire2003}, although in the outer regions of disks and in dwarf\ngalaxies $h_z$ may be higher \\citep[up to $\\sim300$~pc;][]{Walterbrinks1999}.\nWe also assume a constant magnetic field $B=6\\;\\mu{\\rm G}$ as a typical\ngalactic magnetic field \\citep{Beck1996, Heiles2005}. Taking the turbulent\ndecay timescale again to be $\\tau_{\\rm MRI}\n\\simeq9.8\\;(\\lambda_{100}\/\\sigma_{10})\\;\\rm Myr$ we can estimate the driving\nscale $\\lambda=\\lambda_c$.  The critical wavelength for the fastest MRI growth\n\\citep{Balbus1998}\n\\begin{equation}\n\\lambda_c=2\\pi\\:v_A\\; \n\\left[ -\\frac{3+\\alpha}{4}\\;\\frac{d\\Omega^2}{d\\ln r}\\right]^{-1\/2},\n\\end{equation}\nwhere $\\alpha\\equiv d\\ln v\/d\\ln r$, and the Alfv\\'en velocity $v_A^2 = B^2 \/ 4\n\\pi \\rho$.  For a typical galactic magnetic field of $B=6\\;\\mu{\\rm G}$ and a\ndensity of $2\\times10^{-24}$~g~cm$^{-3}$, $\\lambda_c\\sim10^2$~pc.\n\n\nIn Fig.~\\ref{fig:Ek_vs_r}, we compare the energy produced by MRI within a\ndecay time of turbulence, $\\epsilon_{\\rm MRI}\\,\\dot{E}_{\\rm MRI}\\,\\tau_{\\rm\n  MRI}$, with the observed kinetic energy $E_k$ as a function of radius for\ndifferent values of the MRI efficiency. Note that although both $\\dot{E}_{\\rm\n  MRI}$ and $\\tau_{\\rm MRI}$ are functions of radius, the MRI energy input\nonly decreases with radius slowly, not exponentially as $E_k$. In\nFig.~\\ref{fig:Ek_vs_r}, we show a comparison between the SN energy\n(\\S~\\ref{sec:supernova}) and the MRI energy plotted as a function of radius,\nand indicated with green and blue solid lines, respectively. We also plot in\nthe pixel-by-pixel $E_k$ vs.\\ \\sfr\\ plot of Figure~\\ref{fig:Ek_vs_sfr}\\ the\naverage value of the MRI for pixels with each \\sfr, for $\\epsilon_{\\rm\n  MRI}=1$.\n    \nWhile SNe can be identified as the dominant source of energy at $\\Sigma_{\\rm\n  SFR}>-9\\;{\\rm M_\\odot\\;yr^{-1}\\;pc^{-2}}$, the MRI contribution becomes\nimportant at $\\Sigma_{\\rm SFR}<-9\\;{\\rm M_\\odot\\;yr^{-1}\\;pc^{-2}}$ and\ndominant at large galactocentric radii. On the other hand, the kinetic energy\nof turbulence within $r_{25}$ is much higher than the predicted MRI energy,\nindicating that MRI can not account for the observed gas turbulence in regions\nof active star formation.\n\nThe observed $E_k$ is higher than the SN energy for the dwarf galaxies\nHolmberg~II and, more significantly, IC~2574. In these two cases, the gas\nturbulence cannot be explained by SNe \\citep[cf.][]{Stanimirovic2001, Dib2005,\n  Pasquali2008}; still, the MRI could account for the observed regime of\nturbulence.  Our analysis suggests that MRI could dominate regions of low SFR,\nand rules out MRI as an effective turbulence driving mechanism at high SFR.\n\n\n\\subsection{Robustness  of the Approach}\\label{sec:errors}\n\nThree major sources of uncertainty enter our analysis.  The first and largest\nof these are the empirical conversions from UV and IR emission to \\sfr, which\nmay introduce up to a 40\\% uncertainty (\\S~\\ref{sec:limitations}). The\nconversion from \\sfr\\ to SN rate relying on a universal IMF introduces an\nadditional 30\\% uncertainty (\\S~\\ref{sec:supernova}).  A second uncertainty\nenters from the reliance on numerical simulation to evaluate the driving scale\nof SN-driven turbulence and thus the dissipation timescale $\\tau_{\\rm SN}$.\nThe simulations give a 30\\% error for their estimate of the driving scale\n\\citep{Joung2006, Avillez2007}.  The dissipation timescale also depends on the\nmean velocity dispersion, which is, of course, not constant as assumed in our\nestimate. A third uncertainty enters in our estimates of MRI energy, where we\nassume a constant magnetic field and vertical thickness of the disk for all\nthe galaxies of the sample, which may not be the case. The vertical thickness\nof the gaseous component in galaxies increases outwards and may double the\nvalue assumed in our analysis of $h_z=100$~pc. The typical strength of\nmagnetic fields observed in the Milky Way and other galaxies declines slowly\nas a function of galactocentric distance, although the variations of the\nmagnetic field can be much larger azimuthally than radially\n\\citep{Fletcher2004, Han2006, Beck2007}.  These three sources of errors could\nin principle explain those data points lying near $\\epsilon_{\\rm SN}\\gtrsim1$\nand $\\epsilon_{\\rm MRI}\\gtrsim1$.\n\nAside from the potential sources of uncertainties, if we were to interpret our\nempirical finding by taking into account only SNe explosions and MRI, still a\nminor part of the observed turbulence would require $\\epsilon_{\\rm SN}$ and\n$\\epsilon_{\\rm MRI}$ efficiencies uncomfortably high, as high as 100\\%.\nTherefore, we do not exclude that other mechanisms could be efficiently\ndriving some turbulence in the gas. Potentially, within the star forming\nradius ($\\lesssim r_{25}$) stellar winds could be the most effective, while in\nthe outermost regions of \\hi\\ disks, i.e. $r\\gtrsim2\\times r_{25}$, where the\nSNe and the star formation effects are not likely to produce feedback, a floor\nlevel of \\hi\\ velocity dispersion of $\\sim6$~\\kms\\ could be attributed to\nthermal broadening.  Observation of the \\hi\\ in absorption at large\ngalactocentric radii might ultimately help in determining whether the gas is\ncold and turbulent or warm \\citep[see][]{Dickey1993}.\n\n\n\n\\subsection{Is \\shi\\  Controlling the $E_k$ vs \\sfr\\ Relation?}\\label{sec:controlling}\n\nThe gas kinetic energy, $E_k$, is the correct physical quantity to study the\nenergy balance in the ISM. However, gas surface density and SFR are well known\nto correlate \\citep[e.g.][]{Kennicutt1998, Bigiel2008}, so one may wonder\nwhether this drives our observed correlation between $E_k$ and \\sfr.  In other\nwords, the observed covariation between $E_k$ and \\sfr\\ might result from the\ntwo facts: at higher gas mass density, galaxies form stars at higher rate, and\nhigher gas mass bears higher kinetic energy. In order to verify that the\ncorrelation between $E_k$ and \\sfr\\ is not controlled by \\shi, we remove the\neffect of \\shi\\ on the $E_k$ vs \\sfr\\ relation by calculating the partial\ncorrelation coefficient:\n\\begin{equation}\\label{eq:partcorr}\n\\rho_{12.3} = \\frac { \\rho_{12} - \\rho_{13} \\rho_{23} }\n {\\sqrt{ (1-\\rho^2_{13}) (1-\\rho^2_{23}) }},\n\\end{equation}\nwhere $\\rho_{12.3}$ is the partial correlation between $x_1\\equiv\\log\n\\Sigma_{\\rm SFR}$ and $x_2\\equiv\\log E_k$ while controlling for $x_3\\equiv\\log\n\\Sigma_{\\rm HI}$, and $\\rho_{ij}$ is the Pearson's correlation coefficient\nbetween two data sets $x_i$ and $x_j$. Here, considering the quantities in\nlogarithmic scale is convenient as they are correlated as power laws.\nRetaining only data points above a fiducial value for the \\hi\\ mass density,\ni.e.  $\\Sigma_{\\rm HI}\\ge 3$~\\msun\\ \\citep[cf.][]{Bigiel2008}, we obtain the\nvalues listed in Table~\\ref{tab:corrs} for our sample galaxies.  If the\ncorrelation between $E_k$ and \\sfr\\ were completely controlled by \\shi\\ we\nwould expect $\\rho_{12.3} = 0$.  Yet, we find that $0.2\\le\\rho_{12.3}\\le0.6$\nfor our sample galaxies, indicating that the correlation between $E_k$ and\n\\sfr\\ is real.  Equivalently, we show that the correlation $E_k$ vs \\sfr\\ at\nconstant \\shi\\ holds a positive slope in Fig.\\ref{fig:Ek_res_vs_sfr}, in which\nwe remove the contribution from \\shi\\ to $E_k$ by subtracting the average\n$\\avg{E_k}_{\\Sigma \\rm HI}$ within bins of \\shi. The residuals $E_k\n-\\avg{E_k}_{\\Sigma \\rm HI}$ vs \\sfr\\ exhibit a positive correlation,\nindicating that it is not \\shi\\ alone that determines the observed $E_k$ vs\n\\sfr\\ correlation.  Positive slopes in Fig.~\\ref{fig:Ek_res_vs_sfr} and\npositive partial correlation coefficients imply that there is a real, physical\nrelationship between $E_k$ and SFR even at fixed \\shi. The relatively weak\nslopes in Fig.~\\ref{fig:Ek_res_vs_sfr} indicate that higher gas mass density\ncorrelates indeed with higher kinetic energy, simply because it generates more\nstar formation.  Fig.~\\ref{fig:Ek_res_vs_sfr}\\ and the partial correlation\ncoefficients allow us to detect a relationship between $E_k$ and SFR that is\nindependent of \\shi. Comparing Fig.~\\ref{fig:Ek_vs_sfr}\\ and\nFig.~\\ref{fig:Ek_res_vs_sfr}, however, it is clear that most of the\ncorrelation between $E_k$ and SFR in Fig.~\\ref{fig:Ek_res_vs_sfr} closely\ninvolves \\shi\\ (the distributions in Fig.~\\ref{fig:Ek_res_vs_sfr}\\ are very\nflat compared to those in Fig.~\\ref{fig:Ek_vs_sfr}). The basic effect seems to\nbe that higher \\shi\\ results in higher SFR, which creates more $E_k$.  Also,\nnote that if the turbulence, as traced by $E_k$, were effectively suppressing\nstar formation, we would have observed a negative correlation here.\n\n\n\\subsection{Does Turbulence Drive Stochastic Star Formation?}\n\nThe data analysis suggests that the SFR drives the \\hi\\ turbulence through SN\nfeedback. However, the observed correlation could also be interpreted in the\nopposite logical direction, i.e., that the turbulence is driving star\nformation. In fact, as it has been argued \\citep{maclow2004}, turbulence in\nthe ISM has a dual role: (1) to quench star formation by providing pressure\nsupport to the ISM and preventing collapse, and (2) to promote star formation\nby generating stochastic super-critical density enhancements. If the\nturbulence were to drive substantial stochastic star formation, we would\nindeed expect a positive correlation between \\sighi\\ and \\sfr. However,\nFig.~\\ref{fig:m2_vs_r_fit}\\ shows that \\sighi\\ and \\sfr\\ occupy quite\ndifferent dynamic ranges.  While the \\sfr\\ ranges over several orders of\nmagnitude, \\sighi\\ ranges from $\\sim20$ to $\\sim5$~\\kms\\ and is characterized\nby large azimuthal variations.  Although \\sfr\\ positively correlates with\n\\sighi\\ on galactic scales, the large azimuthal variations imply that \\sfr\\ is\nnot well defined for any given value of \\sighi.  This does not preclude\nturbulent induction of star formation in individual regions, but does suggest\nthat this process does not dominate over large scales.  The physical\nexplanation might be that supercritical density fluctuations are often\ndispersed on timescales shorter than the free-fall time, arresting the\ncollapse \\citep{Klessen2000, Elmegreen2002, Joung2006}.\n\n\\section{Effects of Spiral Arm Kinematics and Tidal Interactions}\n\n\nIn the following we discuss other possible mechanisms to produce ISM\nturbulence, such as  spiral arm\nkinematics, tidal interactions, and streamers.\n\nTable~\\ref{tab:objs}, which lists the morphological types of the galaxies of\nour sample, shows that there is no evident trend of the mean velocity\ndispersion, $\\avg{\\sigma}$, among individual galaxies or morphological type.\nSpiral galaxies with strong spiral pattern, e.g. NGC~628 and NGC~3184, have\nsimilar values of typical \\hi\\ velocity dispersion as galaxies with no clear\nspiral structure, e.g. Holmberg~II and IC~2574. Since the spiral arm strength\nshould vary within the sample, we argue that the spiral arm kinematics in our\nsample galaxies are not an important effect in driving turbulence into the\nISM.\n\nAlthough the galaxies Holmberg~II and IC~2574 belong to the M81 \ngroup, they do not show signatures of tidal distortion. In our sample, only NGC~5194 is an interacting galaxy. The \\hi\\ velocity dispersion\nin NGC~5194 is significantly higher than the average for the galaxies in the\nsample (see Fig.~\\ref{fig:s25_avg}).  On the basis of our results, we\nspeculate that  the tidal interaction with the companion NGC~5195\n enhanced the SFR in the disk of NGC~5194, which has\nconsequently driven the velocity dispersion in the \\hi\\ gas to higher values.\n\n\nExtended streamers characterize the galaxies NCG~4736 and NGC~5055.  Their\nradial profiles of the \\hi\\ velocity dispersion exhibit a local increase\noutside the radius of active star formation, i.e., at $r\\sim4'$ ($\\sim\nr_{25}$) for NCG~4736 and at $r\\sim9'$ ($\\sim1.5\\, r_{25}$) for NGC~5055.\nHowever, these local peaks in \\sighi($r$) and the streamers have different\ngalactocentric locations, corresponding to $r\\sim8'$ and $r>11'$ for NCG~4736\nand NGC~5055, respectively. Therefore, we argue that the presence of extended\nstreamers is not likely to be connected to higher \\hi\\ velocity dispersion.\n\n\n\\section{CONCLUSIONS}\n\nCombining high quality maps of \\hi\\ column density and line width provided by\nTHINGS for a sample of dwarf and spiral galaxies, we obtain the following\nresults.\n\\begin{enumerate}\n\\item The \\hi\\ velocity dispersion, \\sighi, declines uniformly as a function\n  of galactocentric distance in all analyzed galaxies.\n  \n\\item At $r_{25}$, the edge of the star-forming region, the \\hi\\ velocity\n  dispersion $\\sigma(r_{25})\\simeq10\\pm2$~\\kms, which is consistent with the\n  mass-weighted median \\hi\\ velocity dispersion $\\avg{\\sigma}$. These findings\n  are independent of the dynamical mass of the galaxy and of their\n  morphological type.\n  \n\\item Within the radius of active star formation ($r\\le r_{25}$), the\n  estimated SN rate and the corresponding energy input rate are sufficient to\n  account for the bulk of observed kinetic energy of turbulence.  For those\n  galaxies of the sample with considerable H$_2$ gas, the SNe can well account\n  for the combined \\hi\\ and H$_2$ gas turbulence.  In this region, the\n  observed instantaneous kinetic energy of the \\hi\\ gas is consistent with the\n  balance between the energy input from the total number of SNe calculated\n  from the observed SFR and the turbulent dissipation predicted by numerical\n  models.  The proportionality between gas $E_k$ and SN energy input rate\n  derived from the SFR provides direct evidence that \\hi\\ turbulence comes\n  from SNe in regions of active SFR. The resulting SN feedback efficiencies\n  are typically $\\epsilon_{\\rm SN}\\times(10^7\\;{\\rm yr}\/\\tau_D)\\simeq0.1$ at\n  SFR levels $\\Sigma_{\\rm SFR} >10^{-9}$~M$_{\\odot}$~yr$^{-1}$~pc$^{-2}$, with\n  the dissipation timescale of turbulence $\\tau_D\\simeq10^7$~yr.\n  \n\\item Within the star forming disk ($r\\le r_{25}$), neither thermal broadening\n  nor MRI can produce the observed \\hi\\ velocity dispersion.  At low SFR,\n  $\\Sigma_{\\rm SFR} <10^{-9}$~M$_{\\odot}$~yr$^{-1}$~pc$^{-2}$, corresponding\n  to large radial distances ($r>r_{25}$), an additional mechanism driving the\n  \\hi\\ velocity dispersion is required to avoid SN efficiencies $\\epsilon_{\\rm\n    SN} > 1$.\n  \n\\item The thermal broadening of the spectral lines, associated to a\n  temperature of $\\sim5000$~K, may be able to explain the observed\n  $\\sigma_{\\rm HI}\\sim 6$~\\kms\\ in the outermost regions of \\hi\\ disks in our\n  sample galaxies, if the required UV radiation to maintain these temperatures\n  is present.  The energy input from MRI can account for the kinetic energy\n  observed in regions of low SFR, $\\Sigma_{\\rm SFR}\n  <10^{-9}$~M$_{\\odot}$~yr$^{-1}$~pc$^{-2}$, at large galactocentric\n  distances.\n  \n\\item We can not unambiguously separate the temperature of the warm and\n  non-turbulent neutral medium from the effect of MRI stirring the ISM.  Both\n  mechanisms are equivalently plausible drivers of the \\hi\\ velocity\n  dispersion observed in the outer parts ($r>r_{25}$) of galaxy disks. We\n  suggest that testing the \\hi\\ line profiles of the gas against a bright\n  background source could ultimately clarify whether the gas in regions of\n  weak star formation is uniformly warm, or contains a cold, turbulent phase,\n  presumably stirred by MRI.  If the gas is actually turbulent, the gas\n  kinetic energy for both high and low star forming regions is consistent in\n  all cases with realistic values of $\\epsilon_{\\rm SN}$ and $\\epsilon_{\\rm\n    MRI}$ efficiencies, suggesting that the feedback provided by both SN\n  explosions and MRI is sufficient to drive the bulk of the observed \\hi\\ \n  turbulence.\n\n\\end{enumerate}\n\n\n\\subsubsection*{Acknowledgments}\n\nWe are grateful to the anonymous referee for the valuable and\ninteresting comments that improved the quality of the paper.\nWe acknowledge helpful discussions with E. Bell, S. Dib, N.\nDziourkevitch, R. Klessen, and A. Pasquali.  M-MML thanks the\nMax-Planck-Gesellschaft for support during his visit to the MPIA.\n\n\\renewcommand{\\baselinestretch}{1}\\normalsize\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStars form via the gravitational collapse of $\\lesssim 0.1$\\,pc dense cores, which are embedded within molecular clouds \\citep{Andre00,Ward-Thompson07,DiFrancesco07}. Prestellar cores become unstable and collapse due to their own gravitational potential. One or several stellar embryos form in their center. This is the beginning of the main accretion phase called the protostellar phase.\nObservations of the molecular line emission from large samples of cores in close star-forming regions revealed that velocity gradients are ubiquitous to prestellar structures at scales of 0.1$-$0.5~pc  \\citep{Goodman93,Caselli02}. These were interpreted as slow rotation inherited from their formation process \\citep{Goodman93,Caselli02, Ohashi99, Redman04, Williams06}. Assuming these gradients trace organized rotational motions, the observed velocities lead to a typical angular rotation velocity of $\\Omega \\sim$ 2~km~s$^{-1}$~pc$^{-1}$ and specific angular momentum values of $j=\\rm{v} \\times r \\sim 10^{-3} - 10^{-1}$~km~s$^{-1}$~pc. \n\n\nDuring the collapse, if the angular momentum of the parent prestellar cores is totally transferred to the stellar embryo, the gravitational force can not counteract the centrifugal force and the stellar embryo fragments prematurely before reaching the main sequence. This is the angular momentum problem for star formation \\citep{Bodenheimer95}.\nAlthough observational studies suggest a trend of decreasing specific angular momentum toward smaller core sizes of $j \\propto r^{1.6}$, the $j$ measured in prestellar structures at scales of 10000~au ($\\sim 10^{-3}$ km~s$^{-1}$~pc, \\citealt{Caselli02}) is still typically three orders of magnitude higher than the one associated with the maximum rotational energy that a solar-type star can sustain ($j_\\mathrm{break} \\sim$ 10$^{18}$ cm$^{2}$~s$^{-1}$ $\\sim$ 3 $\\times$10$^{-6}$~km~s$^{-1}$~pc). The physical mechanisms responsible for the angular momentum redistribution before the matter is accreted by the central stellar object have still to be identified.\n\n\nDuring the star formation process, disk formation is expected to be a consequence of angular momentum conservation during the collapse of rotating cores \\citep{Cassen81, Terebey84}. From observational studies, disks are common in Class~II objects \\citep{Andrews09, Isella09, Ricci10, Spezzi13, Pietu14, Cieza19}.\nThus, disk formation has been naturally considered as a possible solution to the angular momentum problem by redistributing the four orders of magnitude of $j$ measured from prestellar cores to the T-Tauri stars ($j \\sim$2 $\\times$10$^{-7}$~km~s$^{-1}$~pc, \\citealt{Bouvier93}): the disk would store and evacuate the angular momentum of the matter by viscous friction \\citep{Lynden-Bell74, Hartmann98, Najita18} or thanks to disk winds \\citep{Blandford82,Pelletier92,Pudritz07} before the matter is accreted by the central stellar object. However, the spatial distribution of angular momentum during disk formation within star-forming structures at scales between the outer core radius and the stellar surface are still largely unconstrained.\n\n\nClass~0 protostars are the first (proto)stellar objects observed after the collapse in prestellar cores \\citep{Andre93, Andre00}. Due to their youth, most of their mass is still in the form of a dense, collapsing, reservoir envelope surrounding the central stellar embryo (M$_\\mathrm{env} \\gg$ M$_{\\star}$). Thus, they are likely to retain the initial conditions inherited from prestellar cores, in particular regarding angular momentum. \nThe young stellar embryo mass increases via the accretion of the gaseous and dusty envelope in a short timescale ($t<$10$^5$ yr, \\citealt{Evans09,Maury11}). During this main accretion phase, most of the final stellar mass is accreted and, at the same time, the infalling gas must redistribute most of its initial angular momentum before reaching the central stellar embryo. Class~0 protostars are therefore key objects to understand the distribution of angular momentum of the material directly involved in the star formation process and constrain physical mechanisms responsible for the redistribution as disk formation.\n\n\n\n\nClear signatures of rotation \\citep{Belloche02, Belloche04, Chen07} and infalling gas are generally detected in the envelopes of Class~0 protostars (see the review by \\citealt{Ward-Thompson07}). \nThanks to observations of the dense molecular gas emission, rotational motions where characterized in seven Class~0 or I protostellar envelopes at scales between 3500 and 10000~au \\citep{Ohashi97,Belloche02,Chen07}. These envelopes exhibit an average angular momentum of $\\sim$10$^{-3}$\\,km~s$^{-1}$~pc at scales of $r<$5000~au, consistent with the $j$ measured in prestellar cores by \\cite{Caselli02} ($\\sim 10^{-3}$ km~s$^{-1}$~pc). These studies suggest an angular momentum that is constant with radius in Class~0 protostellar envelopes. These flat profiles are generally interpreted as the conservation of the angular momentum. \nFrom hydrodynamical simulations, the conservation of these typical values of angular momentum results in the formation of large rotationally supported disks with radii $>$100~au in a few thousand years \\citep{Yorke99}. However, from observational studies, large disks are rare around Class~0 protostars (r$<$100~au, \\citealt{Maury10, Kurono13, Yen13, Segura-Cox16, Maury18}).\nOnly recent numerical simulations including magnetohydrodynamics and non-ideal effects, such as ambipolar diffusion, Ohmic dissipation, or the Hall effect, allow small rotationally supported disks to be formed ($<$100~au; \\citealt{Machida14, Tsukamoto15, Masson16}).\n\n\nVery few studies have been able to produce resolved profiles of angular momentum to characterize the actual amount of angular momentum present at the smallest scales ($r \\lesssim$1000~au) within protostellar envelopes. From interferometric observations, \\cite{Yen15b} derive specific angular momentum values of $\\sim$2 $\\times$10$^{-4}$~km~s$^{-1}$~pc in seven Class~0 envelopes at scales of $r \\sim$1000~au, values which are below the trend observed by \\cite{Ohashi97}. These studies have put constraints on the angular momentum properties of Class~0 protostellar envelopes and suggest the material at $r \\sim$1000~au must reduce its angular momentum by at least one order of magnitude from outer envelope to disk scales. \\citet{Yen11,Yen17} show specific angular momentum profiles down to $\\sim$350~au in two Class~0 protostellar envelopes: $j \\sim$ 6 $\\times$10$^{-4}$~km~s$^{-1}$~pc at $r \\sim$1000~au and $j \\lesssim$10$^{-4}$~km~s$^{-1}$~pc at $r \\sim$350~au. In this case, conservation of angular momentum during rotating protostellar collapse might not be the dominant process leading to the formation of disks and stellar multiple systems. It is therefore crucial to obtain robust estimates of the angular momentum of the infalling material in protostellar envelopes during the main accretion phase by analyzing the kinematics from the outer regions of the envelope (10000~au, \\citealt{Motte01}) to the protostellar disk ($<$50~au, \\citealt{Maury18}).\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The CALYPSO survey}\nThe Continuum And Lines in Young ProtoStellar Objects (CALYPSO\\footnote{See \\url{http:\/\/irfu.cea.fr\/Projets\/Calypso\/} and \\url{http:\/\/www.iram-institute.org\/EN\/content-page-317-7-158-240-317-0.html}}) IRAM Large Program is a survey of 16 nearby Class 0 protostars (d$<$450~pc), carried out with the IRAM Plateau de Bure interferometer (PdBI) and IRAM 30-meter telescope (30m) at wavelengths of 1.29, 1.37, and 3.18~mm.\nThe CALYPSO sources are among the youngest known solar-type Class~0 objects \\citep{Andre00} with envelope masses of $M_\\mathrm{env} \\sim $1.5~$M_{\\odot}$ and internal luminosities of $L_\\mathrm{bol} \\sim$0.1$-$30~$L_{\\odot}$ \\citep{Maury18}.\n\nThe CALYPSO program allows us to study in detail the Class~0 envelope chemistry \\citep{Maury14,  Anderl16, Simone17,Belloche20}, disk properties \\citep{Maret14,  Maury18, Maret20} and protostellar jets (\\citealt{Codella14, Santangelo15, Podio16, Lefevre17}; Anderl et al. subm.; Podio et al. in prep.).\nOne of the main goals of this large observing program is to understand how the circumstellar envelope is accreted onto the central protostellar object during the Class~0 phase, and ultimately tackle the angular momentum problem of star formation. This paper presents an analysis of envelope kinematics, for the 12 sources from the CALYPSO sample located at $d \\leq$350~pc (see Table \\ref{table:sample}) and discuss our results on the properties of the angular momentum in Class~0 protostellar envelopes. \n\n\n\nWe adopt the dust continuum peak at 1.3~mm (225~GHz) determined from the PdBI datasets by \\cite{Maury18} as origin of the coordinate offsets of the protostellar envelopes (see Table \\ref{table:sample}). We report for each source in Table \\ref{table:sample} the outflow axis considered as the rotation axis and estimated by Podio \\& CALYPSO (in prep.) from high-velocity emission of $^{12}$CO, SiO, and SO at scales $<$10\\hbox{$^{\\prime\\prime}$}. We assume the equatorial axis of the protostellar envelopes, namely the intersection of the equatorial plane with the plane of the sky at the distance of the source, to be perpendicular to the rotation axis.\nThe SiO emission in the CALYPSO maps is very collimated, so the uncertainties on the direction of the rotation axis, and thus on the direction of the equatorial axis, are smaller than $\\pm$10$^{\\circ}$ (Podio \\& CALYPSO, in prep.). For L1521-F, IRAM04191, and GF9-2, no collimated SiO jet is detected, thus, the uncertainties are a bit larger ($\\pm$20$^{\\circ}$). We also use estimates of the inclination of the equatorial plane with respect to the line of sight from the literature. These estimates, which come from geometric models that best reproduce the outflow kinematics observed in molecular emission, are highly uncertain since we do not have access to the 3D-structure of each source.\n\n\n\n\\begin{table*}[!ht]\n\\caption{Sample of CALYPSO Class~0 protostars considered for this analysis.}\n\\begin{center}\n\\resizebox{\\hsize}{!}{\\input{tables\/Table-sample-CALYPSO}}\n\\end{center}\n\\tablefoot{\n\\tablefoottext{a}{Name of the protostars with the multiple components resolved by the 1.3~mm continuum emission from PdBI observations \\citep{Maury18}.}\n\\tablefoottext{b}{Coordinates of the continuum emission peak at 1.3~mm from \\cite{Maury18}.}\n\\tablefoottext{c}{Distance assumed for the individual sources. We adopt a value of 140~pc for the Taurus distance estimated from a VLBA measurement \\citep{Torres09}. The distances of Perseus and Cepheus are taken following recent Gaia parallax measurements that have determined a distance of (293 $\\pm$ 20)~pc \\citep{OrtizLeon18} and (352 $\\pm$ 18)~pc \\citep{Zucker19}, respectively. We adopt a value of 200~pc for the GF9-2 cloud distance \\citep{Wiesemeyer97, Wiesemeyer98} but this distance is very uncertain and some studies estimated a higher distance between 440-470~pc (\\citealt{Viotti69}, C. Zucker, priv. comm.) and 900~pc \\citep{Reid16}.\n}\n\\tablefoottext{d}{Internal luminosities which come from the analysis of \\textit{Herschel} maps from the Gould Belt survey (HGBS, \\citealt{Andre10} and Ladjelate et al. in prep.) and corrected by the assumed distance.}\n\\tablefoottext{e}{Envelope mass corrected by the assumed distance.}\n\\tablefoottext{f}{Outer radius of the individual protostellar envelope determined from dust continuum emission, corrected by the assumed distance. We adopt the radius from PdBI dust continuum emission \\citep{Maury18} when we do not have any information on the 30m continuum from \\cite{Motte01} and for IRAS4A which is known to be embedded into a compressing cloud \\citep{Belloche06}.}\n\\tablefoottext{g}{Position angle of the blue lobe of the outflows estimated from CALYPSO PdBI $^{12}$CO and SiO emission maps (Podio \\& CALYPSO, in prep.). PA is defined east from north. Sources indicated with} \\tablefoottext{$\\star$}{have an asymmetric outflow and the position angles of both lobes are reported. For IRAS2A, IRAS4A, and L1157, previous works done by \\cite{Codella14-bis}, \\cite{Santangelo15}, and \\cite{Podio16}, respectively, show a detailed CALYPSO view of the jets. For L1521F, we use the PA estimated by \\cite{Tokuda14, Tokuda16}.}\n\\tablefoottext{h}{Inclination angle of the equatorial plane with respect to the line of sight. Sources indicated with} \\tablefoottext{$\\star \\star$}{have an inclination angle not well constrained, so we assumed a default value of (30 $\\pm$ 20)$^{\\circ}$.} \n\\tablefoottext{i}{References for the protostar discovery paper, the envelope mass, the envelope radius and then the inclination are reported here.}\n}\n\\tablebib{(1) \\cite{Olinger99}; (2) \\cite{Enoch09}; (3) \\cite{Maury18}; (4) \\cite{Tobin07}; (5) \\cite{Curiel90}; (6) \\cite{Sadavoy14}; (7) \\cite{Motte01}; (8) \\cite{Kwon06}; (9) \\cite{Anglada89}; (10) \\cite{Bachiller95}; (11) \\cite{Girart01}; (12) \\cite{Jennings87}; (13) \\cite{Karska13}; (14) \\cite{Codella04}; (15) \\cite{Maret14}; (16) \\cite{Grossman87}; (17) \\cite{Chini97}; (18) \\cite{Ching16}; (19) \\cite{Desmurs09}; (20) \\cite{Andre99}; (21) \\cite{Andre00}; (22) \\cite{Belloche02}; (23) \\cite{Mizuno94}; (24) \\cite{Tokuda16}; (25) \\cite{Terebey09}; (26) \\cite{Ladd91}; (27) \\cite{Umemoto92}; (28) \\cite{Gueth96}; (29) \\cite{Bachiller01}; (30) \\cite{Schneider79}; (31) \\cite{Wiesemeyer97}.\n}\n\\label{table:sample}\n\\end{table*}\n\n\\section{Observations and dataset reduction}\n\nTo probe the dense gas in our sample of protostellar envelopes, we use high spectral resolution observations of the emission of two molecular lines, C$^{18}$O (2$-$1) at 219.560~GHz and N$_{2}$H$^{+}$ (1$-$0) at 93.171~GHz. In this section, we describe the dataset\\footnote{The datasets used in this paper are available at \\url{http:\/\/www.iram.fr\/ILPA\/LP010\/}.} properties exploited to characterize the kinematics of the envelopes at radii between $r\\sim$50 and 5000~au from the central object. \n\n\\subsection{Observations with the IRAM Plateau de Bure Interferometer}\nObservations of the 12 protostellar envelopes considered here were carried out with the IRAM Plateau de Bure Interferometer (PdBI) between September 2010 and March 2013.\nWe used the 6-antenna array in two configurations (A and C), providing baselines ranging from 16 to 760~m, to carry out observations of the dust continuum emission and a dozen molecular lines, using three spectral setups (around 94~GHz, 219~GHz, and 231~GHz).\nGain and flux were calibrated using CLIC which is part of the GILDAS\\footnote{See \\url{http:\/\/www.iram.fr\/IRAMFR\/GILDAS\/} for more information about the GILDAS software \\citep{Pety05}.} software. \nThe details of CALYPSO observations and the calibration carried out are presented in \\citet{Maury18}. The phase self-calibration corrections derived from the continuum emission gain curves, described in \\citet{Maury18}, were also applied to the line visibility dataset (for all sources in the restricted sample studied here except the faintest sources IRAM04191, L1521F, GF9-2, and L1448-2A).\nHere, we focus on the C$^{18}$O (2$-$1) emission line at 219560.3190~MHz and the N$_{2}$H$^{+}$ (1$-$0) emission line at 93176.2595~MHz, observed with high spectral resolution (39~kHz channels, i.e., a spectral resolution of 0.05~km~s$^{-1}$ at 1.3~mm and 0.13~km~s$^{-1}$ at 3~mm).\nThe C$^{18}$O (2$-$1) maps were produced from the continuum-subtracted visibility tables using either (i) a robust weighting of 1 for the brightest sources to minimize the side-lobes, or (ii) a natural weighting for the faintest sources (IRAM04191, L1521F, GF9-2, and L1448-2A) to minimize the rms noise values. We resampled the spectral resolution to 0.2~km~s$^{-1}$ to improve the signal-to-noise ratio of compact emission. \nThe N$_{2}$H$^{+}$ (1$-$0) maps were produced from the continuum-subtracted visibility tables using a natural weighting for all sources. In all cases, deconvolution was carried out using the Hogbom algorithm in the MAPPING program of the GILDAS software.\n\n\n\n\\subsection{Short-spacing observations from the IRAM 30-meter telescope}\nThe short-spacing observations were obtained at the IRAM 30-meter telescope (30m) between November 2011 and November 2014. Details of the observations for each source are reported in Table \\ref{table:temps-observations-30m}. We observed the C$^{18}$O (2$-$1) and N$_{2}$H$^{+}$ (1$-$0) lines using the heterodyne Eight MIxer Receiver (EMIR) in two atmospheric windows: E230 band at 1.3~mm and E090 band at 3~mm \\citep{Carter12}. The Fast Fourier Transform Spectrometer (FTS) and the VErsatile SPectrometer Array (VESPA) were connected to the EMIR receiver in both cases. \nThe FTS200 backend provided a large bandwidth (4~GHz) with a spectral resolution of 200~kHz (0.27~km~s$^{-1}$) for the C$^{18}$O line, while VESPA provided  high spectral resolution observations (20~kHz channel or 0.063~km~s$^{-1}$) of the N$_{2}$H$^{+}$ line.\nWe used the on-the-fly spectral line mapping, with the telescope beam moving at a constant angular velocity to sample regularly the region of interest (1$\\arcmin$ $\\times$ 1$\\arcmin$ coverage for the C$^{18}$O emission and 2$\\arcmin$ $\\times$ 2$\\arcmin$ coverage for the N$_{2}$H$^{+}$ emission).\nThe mean atmospheric opacity at 225 GHz was $\\tau_{225} \\sim 0.2$ during the observations at 1.3~mm and $\\tau_{225} \\sim 0.5$ during the observations at 3~mm.\nThe mean values of atmospheric opacity are reported for each source in Table \\ref{table:temps-observations-30m}.\nThe telescope pointing was checked every 2$-$3~h on quasars close to the CALYPSO sources, and the telescope focus was corrected every 4$-$5~h using the planets available in the sky.\nThe single-dish dataset were reduced using the MIRA and CLASS programs of the GILDAS software following the standard steps: flagging of incorrect channels, temperature calibration, baseline subtraction, and gridding of individual spectra to produce regularly-sampled maps.\n\n\n\\subsection{Combination of the PdBI and 30m data}\nThe IRAM PdBI observations are mostly sensitive to compact emission from the inner envelope.\nInversely, single-dish dataset contains information at envelope scales ($r \\sim 5-40\\hbox{$^{\\prime\\prime}$}$) but its angular resolution does not allow us to characterize the inner envelope emission at scales smaller than the beamwidth. \nTo constrain the kinematics at all relevant scales of the envelope, one has to build high angular resolution dataset which recovers all emission of protostellar envelopes.\nWe merged the PdBI and the 30m datasets (hereafter PdBI+30m) for each tracer using the pseudo-visibility method\\footnote{For details, see \\url{http:\/\/www.iram.fr\/IRAMFR\/GILDAS\/doc\/pdf\/map.pdf}.}: we generated pseudo-visibilities from the Fourier transformed 30m image data, which are then merged to the PdBI dataset in the MAPPING program of the GILDAS software. This process degrades the angular resolution of the PdBI dataset but recovers a large fraction of the extended emission. The spectral resolution of the combined PdBI+30m dataset is limited by the 30m dataset at 1.3~mm (0.27~km~s$^{-1}$) and the PdBI one at 3~mm (0.13~km~s$^{-1}$). \n\n\nWe produced the N$_{2}$H$^{+}$ (1$-$0) combined datacubes in such a way to have a synthesized beam size $<2\\hbox{$^{\\prime\\prime}$}$ and a noise level $<$10~mJy~beam$^{-1}$. As the N$_{2}$H$^{+}$ emission traces preferably the outer protostellar envelope, we used a natural weighting to build the combined maps to minimize the noise rather than to maximize angular resolution.\nThe C$^{18}$O (2$-$1) combined maps were produced using a robust weighting scheme, in order to obtain synthesized beam sizes close to the PdBI ones, and to minimize the side-lobes. In all cases, the deconvolution was carried out using the Hogbom algorithm in MAPPING.\n\n\n\n\n\\subsection{Properties of the analyzed maps}\nFollowing the procedure described above, we have obtained, for each source of the sample, a set of three cubes for each of the two molecular tracers C$^{18}$O (2$-$1) and N$_{2}$H$^{+}$ (1$-$0), probing the emission at different spatial scales (PdBI map, combined PdBI+30m map, and 30m map). \nIn order to build maps with pixels that contain independent dataset and avoid oversampling, we inversed visibilities from the PdBI and the combined PdBI+30m datasets using only 4 pixels per synthesized beam, and we smoothed the resulting maps afterwards to obtain 2 pixels per element of resolution. \nThe properties of the resulting maps are reported in Appendix \\ref{properties-maps}. The spatial resolution of the molecular line emission maps is reported in Tables \\ref{table:beams-c18o} and \\ref{table:beams-n2hp}. \nThe spatial extent of the molecular emission, the rms noise levels, and the integrated fluxes are reported in Tables \\ref{table:details-obs-c18o} and \\ref{table:details-obs-n2hp}.\n\n\n\n\n\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L14482A\/intensity-maps-L1448-2A-mean.pdf}\n\\caption{Integrated intensity maps of N$_{2}$H$^{+}$ (1$-$0) (top) and C$^{18}$O (2$-$1) (bottom) emission from the PdBI (left), combined (middle), and 30m (right) datasets for L1448-2A.\nThe white crosses represent the positions of the binary system determined from the 1.3~mm dust continuum emission.\nThe black cross represents the middle position between the binary system. The clean beam is shown by an ellipse on the bottom left of each map.\nThe black lines represent the integrated intensity contours of each tracer starting at 5$\\sigma$ and increasing in steps of 25$\\sigma$ for N$_{2}$H$^{+}$ and 10$\\sigma$ for C$^{18}$O (see Tables \\ref{table:details-obs-c18o} and \\ref{table:details-obs-n2hp}). Be careful, the spatial scales of the maps are not uniform in all panels.}\n\\label{fig:intensity-maps-L1448-2A}\n\\end{figure*}\n\n\n\n\n\n\\section{Envelope kinematics from high dynamic range datasets}\n\n\n\\subsection{Integrated intensity maps}\n\nTo identify at which scales of the protostellar envelopes the different datasets are sensitive to, we produced integrated intensity maps by integrating spectra of each pixel for the molecular lines C$^{18}$O (2$-$1) and N$_{2}$H$^{+}$ (1$-$0) from the PdBI, combined, and 30m datasets for each source. \nFor C$^{18}$O (2$-$1), we integrated each spectrum on a velocity range of $\\pm$ 2.5~km~s$^{-1}$ around the velocity of the peak of the mean spectrum of each source. The 1$-$0 line of N$_{2}$H$^{+}$ has a hyperfine structure with seven components (see Fig. \\ref{fig:spectra-30m-L1448C}). We integrated the N$_{2}$H$^{+}$ spectra over a range of 20~km~s$^{-1}$ encompassing the seven \ncomponents. Figure~\\ref{fig:intensity-maps-L1448-2A} shows as an example the integrated intensity maps obtained for L1448-2A. The integrated intensity maps of the other sources are provided in Appendix~\\ref{sec:comments-indiv-sources}.\n\nWe used the integrated intensity maps to measure the average emission size of each tracer in each dataset above a 5$\\sigma$ threshold.\nThe values reported in Tables \\ref{table:details-obs-c18o} and \\ref{table:details-obs-n2hp} are the average of two measurements: an intensity cut along the equatorial axis and circular averages at different radii around the intensity peak position of the source. Only pixels whose intensity is at least 5 times higher than the noise in the map are considered to build these intensity profiles. The FWHM of the adjustment by a Gaussian function allows us to determine the average emission size of the sources. For both tracers and for all sources in our sample, the emission is detected above 5$\\sigma$ in an area larger in the combined datasets than in the PdBI datasets, and smaller than in the 30m ones (see Tables \\ref{table:details-obs-c18o} and \\ref{table:details-obs-n2hp}). Our three datasets are thus not sensitive to the same scales and allow us to probe different scales within the 12 sampled protostellar envelopes: the 30m datasets trace the outer envelope, the PdBI datasets the inner part and the combined ones the intermediate scales.\n\n\nThe C$^{18}$O and N$_{2}$H$^{+}$ molecules do not trace the same regions of the protostellar envelope either: \\cite{Anderl16} report from an analysis of the CALYPSO survey that the N$_{2}$H$^{+}$ emission forms a ring around the central C18O emission in four sources.\nPrevious studies \\citep{Bergin02,Maret02,Maret07,Anderl16} show that N$_{2}$H$^{+}$, which is abundant \nin the outer envelope, is chemically destroyed when the temperature in the envelope reaches the critical temperature ($T \\gtrsim$20K) at which CO desorbs from dust ice mantles. Thus, while N$_{2}$H$^{+}$ can be used to probe the envelope kinematics at outer envelope scales, C$^{18}$O can be used as a complementary tracer of the gas kinematics at smaller radii where the embedded protostellar embryo heats the gas to higher temperatures. \n\nThe C$^{18}$O emission is robustly detected ($>$5$\\sigma$) in our PdBI observations for most sources, except for L1521F and IRAM04191 which are the lowest luminosity sources of our sample (see Table \\ref{table:sample}), and for SVS13-B where the emission is dominated by its companion, the Class~I protostar SVS13-A. For most sources, the interferometric map obtained with the PdBI shows mostly compact emission ($r < 3 \\hbox{$^{\\prime\\prime}$}$, see Table \\ref{table:details-obs-c18o}). However, the C$^{18}$O emission from the 30m datasets shows more complex structures (see Appendix~\\ref{sec:comments-indiv-sources}). Assuming that, under the hypothesis of spherical geometry, the emission from a protostellar envelope is compact ($r \\lesssim$40$\\hbox{$^{\\prime\\prime}$}$, i.e., $\\lesssim$10000~au, see Table \\ref{table:sample}) and stands out from the environment in which it is embedded, the 30m emission of L1448-2A, L1448-C, and IRAS4A comes mainly from the envelope.\n\nThe N$_{2}$H$^{+}$ emission is detected in our combined observations for all sources. In the four sources studied by \\cite{Anderl16}, they do not detect the emission at the 1.3~mm continuum peak, but emission rings around the C$^{18}$O central emission. From Table \\ref{table:details-obs-n2hp}, we noticed two types of emission morphologies based on the PdBI dataset: compact ($r <$7$\\hbox{$^{\\prime\\prime}$}$, see Table \\ref{table:details-obs-n2hp}) or filamentary ($r \\geq$9$\\hbox{$^{\\prime\\prime}$}$). In the same way as the C$^{18}$O emission, the N$_{2}$H$^{+}$ emission from the 30m datasets shows complex structures with radius $r \\gtrsim$40$\\hbox{$^{\\prime\\prime}$}$ for most sources, except for five sources (IRAM04191, L1521F, L1448-NB, L1448-C, and L1157) where the emission is consistent with the compact emission of the protostellar envelope.\n\nThe C$^{18}$O emission from the PdBI is not centered on the continuum peak for three sources in our sample: IRAS4A, L1448-NB, and L1448-2A (see Appendix~\\ref{sec:comments-indiv-sources}). \nFor each of these sources, the PdBI 1.3~mm dust continuum emission map resolves a close binary system ($<$600~au) with both components embedded in the same protostellar envelope (\\citealt{Maury18}, see Table \\ref{table:sample}). The origin of the coordinate offsets is chosen to be the main protostar, secondary protostar, and the middle of the binary system for IRAS4A, L1448-NB, and L1448-2A, respectively, to study the kinematics in a symmetrical way.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.3,angle=0,trim=0.3cm 0cm 0.4cm 4cm,clip=true]{fig-L14482A\/mean-spectrum-N2H+-L1448-2A.pdf}\n\\includegraphics[scale=0.3,angle=0,trim=0.3cm 0cm 0.4cm 4cm,clip=true]{fig-L14482A\/mean-spectrum-C18O-L1448-2A.pdf}\n\\caption{Mean spectra of the N$_2$H$^{+}$ (top) and C$^{18}$O (bottom) molecular lines from the 30m datasets for L1448-2A. The best fits of the spectra, by a hyperfine structure and a Gaussian line profile models respectively, are represented in green solid lines. In the top panel, the velocity axis corresponds to the isolated HFS component $1_{01}-0_{12}$. The systemic velocity is estimated to be 4.10~km~s$^{-1}$ for this source (see Table \\ref{table:vitesse-systemique-n2hp-30m}). }\n\\label{fig:spectra-30m-L1448C}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\subsection{Velocity gradients in protostellar envelopes} \\label{sec:velocity-maps} \nTo quantify centroid velocity variations at all scales of the protostellar envelopes, we produced centroid velocity maps of each Class~0 protostellar envelope by fitting all individual spectra (pixel by pixel) by line profile models in the CLASS program of the GILDAS software. We only considered the line intensity detected with a signal-to-noise ratio higher than 5.\nWe fit the spectra to be able to deal with multiple velocity components. Indeed, because protostellar envelopes are embedded in large-scale clouds, multiple velocity components can be expected on some lines of sight where both the protostellar envelope and the cloud emit. For example, \\cite{Belloche06} find several velocity components in their 30m of the N$_2$H$^+$ emission of IRAS4A (see Appendix \\ref{sec:comments-IRAS4A}). \nExcept for IRAS4A and IRAS4B for which we fit two velocity components (see details in Appendix \\ref{sec:comments-indiv-sources}), for most sources we used a Gaussian line profile to model the C$^{18}$O (2$-$1) emission, with the line intensity, full width at half maximum (FWHM), and centroid velocity let as free parameters (see Fig. \\ref{fig:spectra-30m-L1448C}). In the case of N$_{2}$H$^{+}$ (1$-$0), we used a hyperfine structure (HFS) line profile to determine the FWHM and centroid velocity of the molecular line emission (see Fig. \\ref{fig:spectra-30m-L1448C}). Figures~\\ref{fig:velocity-maps-L1448-2A} to \\ref{fig:velocity-maps-GF92} show the centroid velocity maps obtained for each source of the sample using the PdBI, combined, and 30m datasets for both the C$^{18}$O and N$_{2}$H$^{+}$ emission.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L14482A\/velocity-maps-L1448-2A-mean.pdf}\n\\caption{Centroid velocity maps of N$_{2}$H$^{+}$ (1$-$0) (top) and C$^{18}$O (2$-$1) (bottom) emission from the PdBI (left), combined (middle), and 30m (right) datasets for L1448-2A. The blue and red solid arrows represent the directions of the blue- and red-shifted outflow lobes, respectively. The white crosses represent the positions of the binary system determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}).\nThe black cross represents the middle position between the binary system. The clean beam is shown by an ellipse on the bottom left. The integrated intensity contours in black are the same as in Fig. \\ref{fig:intensity-maps-L1448-2A}.\n}\n\\label{fig:velocity-maps-L1448-2A}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1448N\/velocity-maps-L1448-NB2.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for L1448-NB. The white cross represents the position of main protostar L1448-NB1 determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}). The black cross represents the position of the secondary protostar L1448-NB2 of the multiple system.\n}\n\\label{fig:velocity-maps-L1448NB}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1448C\/velocity-maps-L1448C.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for L1448-C. \n}\n\\label{fig:velocity-maps-L1448C}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-IRAS2A\/velocity-maps-IRAS2A.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for IRAS2A. \n}\n\\label{fig:velocity-maps-IRAS2A}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-SVS13B\/velocity-maps-SVS13B.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for SVS13-B. The white cross represents the position of the Class~I protostar SVS13-A determined from the 1.3~mm dust continuum emission \\citep{Maury18}.\n}\n\\label{fig:velocity-maps-SVS13B}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-IRAS4A\/velocity-maps-IRAS4A1.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for IRAS4A. The white cross represents the position of secondary protostar IRAS4A2 determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}). The black cross represents the position of the main protostar IRAS4A1 of the multiple system. \n}\n\\label{fig:velocity-maps-IRAS4A}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-IRAS4B\/velocity-maps-IRAS4B.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for IRAS4B. The white crosses represent the position of secondary protostar IRAS4B2 and the position of IRAS4A, respectively, determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}). The black cross represents the position of the secondary protostar L1448-NB2 of the multiple system.\n}\n\\label{fig:velocity-maps-IRAS4B}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-IRAM04191\/velocity-maps-IRAM04191.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for IRAM04191. The white cross represents the position of the Class~I protostar IRAS04191. The black cross represents the position of IRAM04191 determined from the 1.3~mm dust continuum emission (see Table \\ref{table:sample}).\n}\n\\label{fig:velocity-maps-IRAM04191}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1521F\/velocity-maps-L1521F.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A} for L1521F. The white cross represents the position of the starless dense core MMS-2.\n}\n\\label{fig:velocity-maps-L1521F}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1527\/velocity-maps-L1527.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for L1527. \n}\n\\label{fig:velocity-maps-L1527}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-L1157\/velocity-maps-L1157.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for L1157. \n}\n\\label{fig:velocity-maps-L1157}\n\\end{figure*}\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.5,angle=0,trim=0cm 1.5cm 0cm 1.5cm,clip=true]{fig-GF92\/velocity-maps-GF92.pdf}\n\\caption{Same as Figure \\ref{fig:velocity-maps-L1448-2A}, but for GF9-2. \n}\n\\label{fig:velocity-maps-GF92}\n\\end{figure*}\n\nFor most sources in our sample, these centroid velocity maps reveal organized velocity patterns with blue-shifted and red-shifted velocity components on both sides of the central stellar embryo, along the equatorial axis where such velocity gradients could be due to rotation of the envelopes.\nThe global kinematics in Class~0 envelopes is a complex combination of rotation, infall, and outflow motions. The observed velocities are projected on the line of sight and thus, are a mix of the various gas motions. Therefore, it is not straightforward to interpret a velocity gradient in terms of the underlying physical process producing it.\nIn order to have an indication of the origin of these gradients, we performed a least-square minimization of a linear velocity gradient model on the velocity maps following:\n\\begin{equation}\n\\rm{v}_\\mathrm{grad}=\\rm{v}_0 + a \\Delta \\alpha + b \\Delta \\beta, \n\\label{eq:vgrad}\n\\end{equation}\nwith $\\Delta \\alpha$ and $\\Delta \\beta$ the offsets with respect to the central source \\citep{Goodman93}.\n\n\nThis simple model provides an estimate of the reference velocity $\\mathrm{v}_0$ called systemic velocity, the direction $\\Theta$, and the amplitude $G$ of the mean velocity gradient. One would expect a mean gradient perpendicular to the outflow axis if the velocity gradient was due to rotational motions in an axisymmetric envelope. A mean gradient oriented along the outflow axis could be due to jets and outflows or infall in a flattened geometry. \nThe gradients were fit on the region of the velocity maps shown in Fig. \\ref{fig:velocity-maps-L1448-2A}, namely 10$\\hbox{$^{\\prime\\prime}$} \\times$ 10$\\hbox{$^{\\prime\\prime}$}$ in the PdBI and combined datasets for the C$^{18}$O emission (lower left and central panels), 40$\\hbox{$^{\\prime\\prime}$} \\times$ 40$\\hbox{$^{\\prime\\prime}$}$ for the N$_{2}$H$^{+}$ emission from the PdBI and combined datasets (upper left and central panels), and 80$\\hbox{$^{\\prime\\prime}$} \\times$ 80$\\hbox{$^{\\prime\\prime}$}$ and 160$\\hbox{$^{\\prime\\prime}$} \\times$ 160$\\hbox{$^{\\prime\\prime}$}$, respectively for the C$^{18}$O and N$_{2}$H$^{+}$ emission from the 30m datasets (right panels).\nTable \\ref{table:gradient-velocity-fit} reports for each source the significant mean velocity gradients detected with an amplitude higher than 2$\\sigma$. No significant velocity gradient is observed for IRAM04191, L1521F, and SVS13-B in C$^{18}$O emission at scales of $r<$5\\hbox{$^{\\prime\\prime}$} or for L1448-C and IRAS4B at $r>$30\\hbox{$^{\\prime\\prime}$} (see Table \\ref{table:gradient-velocity-fit}).\n\nSeven of the 12 sources in our sample show a mean gradient in C$^{18}$O emission aligned with the equatorial axis ($\\Delta \\Theta <$30$^{\\circ}$) which could trace rotational motions of the envelope at scales of $r<$5\\hbox{$^{\\prime\\prime}$}. \nAt similar scales, four sources (L1448-NB, L1521F, L1157, GF9-2) show gradients with intermediate orientation (30$^{\\circ}< \\Delta \\Theta <$60$^{\\circ}$). \nFinally, L1448-2A shows a mean gradient aligned to the outflow axis rather than the equatorial axis ($\\Delta \\Theta >$60$^{\\circ}$). For these last five sources, the gradients observed could be due to a combination of rotation, ejection, and infall motions. \nFor all sources, we noticed a systematic dispersion of the direction of the velocity gradient from inner to outer scales in the envelope (see Fig. \\ref{fig:evolution-theta-plot}). We discuss in Sect. \\ref{sec:velocity-gradient-1000au} whether this shift in direction of the velocity gradient is due to a transition from rotation-dominated inner envelope to collapse-dominated outer envelope at $r>$1500~au, or is due to the different molecular tracers used for this analysis.\nIn most sources, the gradient moves away from the equatorial axis as the scale increases. Only three sources (IRAM04191, L1521F, and L1527) show a gradient close to the equatorial axis with $\\Delta \\Theta <$30$^{\\circ}$ at 2000~au in N$_{2}$H$^{+}$ emission from the combined dataset while four sources show a complex gradient and five sources have a $\\Delta \\Theta >$60$^{\\circ}$.\n\n\n\\begin{sidewaystable*}\n\\centering\n\\caption{Estimation of the systemic velocity, the mean velocity gradient amplitude and its orientation from linear gradient fit of centroid velocity maps in C$^{18}$O and N$_{2}$H$^{+}$ emission from the PdBI, the combined, and the 30m datasets for the CALYPSO sample sources.}\n\\label{table:gradient-velocity-fit}\n\\resizebox{1\\textwidth}{!}{\\input{tables\/Table-velocity-gradient-fit}}\n\\tablefoot{ \\tablefoottext{a}{Position angle of the redshifted lobe of the velocity gradient defined from north to east.}\n\\tablefoottext{b}{Absolute value, between 0$^{\\circ}$ and 90$^{\\circ}$, of the difference between the angle of the mean gradient and the angle of the equatorial axis. The equatorial axis is defined perpendicularly to the direction of the outflows (see Table \\ref{table:sample}).}\n\\tablefoottext{$\\star$}{For IRAM04191, the N$_{2}$H$^{+}$ emission in the 30m dataset was fit ignoring the pixels close to the Class~I protostar IRAS04191 in the field of view (see Appendix \\ref{sec:comments-indiv-sources}).}}\n\\end{sidewaystable*}\n\n\n\n\n\n\\subsection{High dynamic range position-velocity diagrams to probe rotational motions} \\label{subsec:diagram-PV-construction}\nTo investigate rotational motions and characterize the angular momentum properties in our sample of Class~0 protostellar envelopes, we build the position-velocity (PV$_\\mathrm{rot}$) diagrams along the equatorial axis. We assumed the position angle of the equatorial axis as orthogonal to the jet axis reported in Table \\ref{table:sample}. \nThe choice of this equatorial axis allows us to maximize sensitivity to rotational motions and minimize potential contamination on the line of sight due to collapsing or outflowing gas \\citep{Yen13}. The velocities reported in the PV$_\\mathrm{rot}$ diagram are corrected for the inclination $i$ of the equatorial plane with respect to the line of sight (see Table \\ref{table:sample}). \nWe note that the correction for inclination is a multiplicative factor, thus if this inclination angle is not correctly estimated, the global observed shape is not distorted.\n\n\nThe analysis described in detail in Appendix \\ref{details-diagram-PV-construction} allows us to build a PV$_\\mathrm{rot}$ diagram with a high dynamic range from 50~au up to 5000~au for each source as follows (see the example of L1527 in Fig. \\ref{fig:PV-diagram-construction}):\\\\\n\\indent $\\bullet$ To constrain the PV$_\\mathrm{rot}$ diagram at the smallest scales resolved by our dataset ($\\sim$0.5$\\hbox{$^{\\prime\\prime}$}$), we use the PdBI C$^{18}$O datasets that we analyze in the (u,v) plane to avoid imaging and deconvolution processes (see label \"C$^{18}$O PdBI\" in Fig. \\ref{fig:PV-diagram-construction}). We only kept central emission positions in the channel maps at a position angle $<$|45$^{\\circ}|$ with respect to the equatorial axis (see Appendix \\ref{details-diagram-PV-construction}). \\\\\n\\indent $\\bullet$ Since the C$^{18}$O extended emission is filtered out by the interferometer, we used the combined C$^{18}$O emission to populate the PV$_\\mathrm{rot}$ diagram at the intermediate scales of the protostellar envelopes (see label \"C$^{18}$O combined\" in Fig. \\ref{fig:PV-diagram-construction}). The C$^{18}$O molecule remains the most precise tracer when the temperature is higher than $\\sim$20K because below, the C$^{18}$O molecule freezes onto dust ice mantles. To determine the transition radius $R_\\mathrm{trans}$ between the two tracers, we calculate the C$^{18}$O and N$_{2}$H$^{+}$ column densities along the equatorial axis from the combined integrated intensity maps (see Appendix \\ref{sec:column-density} and green points in Fig. \\ref{fig:PV-diagram-construction}).\\\\\n\\indent $\\bullet$ At radii $r>R_\\mathrm{trans}$, the N$_{2}$H$^{+}$ emission traces better the envelope dense gas. We use the combined N$_{2}$H$^{+}$ emission maps to analyze the envelope kinematics at larger intermediate scales. When the N$_{2}$H$^{+}$ column density profile reaches a minimum value due to the sensitivity of the combined dataset, this dataset is no longer the better dataset to provide a robust information on the velocity (see label \"N$_{2}$H$^{+}$ combined\" in Fig. \\ref{fig:PV-diagram-construction}). \\\\\n\\indent $\\bullet$ Finally, we use the 30m N$_{2}$H$^{+}$ emission map to populate the PV$_\\mathrm{rot}$ diagram at the largest scales of the envelope (see label \"N$_{2}$H$^{+}$ 30m\" in Fig. \\ref{fig:PV-diagram-construction}).\\\\\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=9cm]{figures-discussion\/schema-construction-PVdiagram.pdf}\n\\caption{Plot summarizing the combination of tracers and datasets used to build high dynamic range PV$_\\mathrm{rot}$ diagrams in the L1527 envelope.\nThe transition radii between the different datasets (PdBI, combined, and 30m) and the two C$^{18}$O and N$_{2}$H$^{+}$ tracers represented by the dashed lines are given in Table \\ref{table:radius-lines}. The green points show the column density profiles along the equatorial axis of C$^{18}$O and N$_{2}$H$^{+}$ estimated from the combined datasets (see Appendix \\ref{sec:column-density}).\n}\n\\label{fig:PV-diagram-construction}\n\\end{figure}\n\nThe CALYPSO datasets allow us to continuously estimate the velocity variations along the equatorial axis in the envelope over scales from 50~au up to 5000~au homogeneously for each protostar. \nFigure~\\ref{fig:PV-diagrams-1} shows the PV$_\\mathrm{rot}$ diagrams built for all sources of the sample. The systemic velocity used in the PV$_\\mathrm{rot}$ diagrams are determined in Appendix \\ref{sec:systemic-velocity}.\n\n\nThe method of building PV$_\\mathrm{rot}$ diagrams described above and in Appendix \\ref{details-diagram-PV-construction} corresponds to an ideal case with a detection of a continuous blue-red velocity gradient along the direction perpendicular to the outflow axis in the velocity maps. In practice, the direction of velocity gradients is not always continuous at all scales probed by our observations (see Table \\ref{table:gradient-velocity-fit} and Figure \\ref{fig:evolution-theta-plot}).\nFor some sources, to constrain the PV$_\\mathrm{rot}$ diagrams, we did not take the kinematic information at all scales of the envelope into account. Velocity gradients can be considered as probing rotational motions if the following criteria are met:\\\\\n$\\indent \\bullet$ We only consider the significant velocity gradients reported in Table \\ref{table:gradient-velocity-fit} with a blue- and a red-shifted velocity components observed on each side of the protostellar embryo, itself at the systemic velocity $\\mathrm{v}_0$. For example, we only take the C$^{18}$O emission from the 30m map into account for L1521F (see Fig. \\ref{fig:velocity-maps-L1521F}). \\\\\n$\\indent \\bullet$ We only take the velocity gradients aligned with the equatorial axis ($\\Delta \\Theta <$60$^{\\circ}$) into account in order to minimize contamination by infall and ejection motions. For example, we do not report in the PV$_\\mathrm{rot}$ diagrams the N$_{2}$H$^{+}$ velocity gradients from the combined maps for L1448-NB, IRAS2A, SVS13-B, and GF9-2 (see Table \\ref{table:gradient-velocity-fit}).\\\\\n$\\indent \\bullet$ We do not consider the discontinuous velocity gradients which show an inversion of the blue- and red-shifted velocity components along the equatorial axis from inner to outer envelope scales. For example, we do not report in the PV$_\\mathrm{rot}$ diagrams the N$_{2}$H$^{+}$ velocity gradients from the 30m maps for IRAM04191 and L1157 (see Figs. \\ref{fig:velocity-maps-IRAM04191} and \\ref{fig:velocity-maps-L1157}). \\\\\n\nWhen velocity gradients with a blue- and a red-shifted velocity components observed on each side of the protostellar embryo are continuous from inner to outer envelope scales but shifted from the equatorial axis ($\\Delta \\Theta \\geq$60$^{\\circ}$), we only report upper limits on rotational velocities in the PV$_\\mathrm{rot}$ diagrams.\nThe sources in our sample show specific individual behaviors, therefore we followed as closely as possible the method of building the PV$_\\mathrm{rot}$ diagram adapting it on a case-by-case basis. \n\n\n \n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=9cm]{fig-L14482A\/overplot-rotation-column-density-log-L1448-2A.pdf}\n\\includegraphics[width=9cm]{fig-L1448N\/overplot-rotation-column-density-log-L1448-NB2.pdf}\n\\includegraphics[width=9cm]{fig-L1448C\/overplot-rotation-column-density-log-L1448C.pdf}\n\\includegraphics[width=9cm]{fig-IRAS2A\/overplot-rotation-column-density-log-IRAS2A.pdf}\n\\includegraphics[width=9cm]{fig-SVS13B\/overplot-rotation-column-density-log-SVS13B.pdf}\n\\includegraphics[width=9cm]{fig-IRAS4A\/overplot-rotation-column-density-log-IRAS4A.pdf}\n\\caption{Position-velocity diagram along the equatorial axis of the CALYPSO protostellar envelopes. Blue and red dots show the blue- and red-shifted velocities, respectively. The arrows display the upper limits of $\\mathrm{v}_\\mathrm{rot}$ determined from velocity maps that do not exhibit a spatial distribution of velocities as organized as one would expect from rotation motions (see Sect. \\ref{subsec:diagram-PV-construction} and Appendix \\ref{details-diagram-PV-construction}). Green dots show the column density profiles along the equatorial axis. Dots and large dots show the C$^{18}$O and N$_{2}$H$^{+}$ data, respectively. The dashed curve shows the best fit with a power-law model leaving the index $\\alpha$ as a free parameter ($\\mathrm{v}_\\mathrm{rot} \\propto r^{\\alpha}$) whereas the dotted curve shows the best fit with a power-law model with a fixed index $\\alpha$=-1. The vertical dashed lines show the transition radii between the different datasets (PdBI, combined, and 30m) and the two tracers as illustrated in Fig. \\ref{fig:PV-diagram-construction} and given in Table \\ref{table:radius-lines}.\n}\n\\label{fig:PV-diagrams-1}\n\\end{figure*}\n\n\n\n\\begin{figure*}[!ht]\n\\addtocounter{figure}{-1}\n\\centering\n\\includegraphics[width=9cm]{fig-IRAS4B\/overplot-rotation-column-density-log-IRAS4B.pdf}\n\\includegraphics[width=9cm]{fig-IRAM04191\/overplot-rotation-column-density-log-IRAM04191.pdf}\n\\includegraphics[width=9cm]{fig-L1521F\/overplot-rotation-column-density-log-L1521F.pdf}\n\\includegraphics[width=9cm]{fig-L1527\/overplot-rotation-column-density-log-L1527.pdf}\n\\includegraphics[width=9cm]{fig-L1157\/overplot-rotation-column-density-log-L1157.pdf}\n\\includegraphics[width=9cm]{fig-GF92\/overplot-rotation-column-density-log-GF92.pdf}\n\\caption{Continued.\n}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\nIn this section, we discuss the presence of rotation in the protostellar envelopes from the PV$_\\mathrm{rot}$ diagrams (see Sect. \\ref{sec:results-PV-diagram} and Fig. \\ref{fig:PV-diagrams-1}). We build the distribution of specific angular momentum associated with the PV$_\\mathrm{rot}$ diagrams (see Sect. \\ref{subsec:distribution-j}) and explore the possible solutions to explain the $j(r)$ profiles observed in the inner ($r<$1600~au, see Sect. \\ref{sec:j-inner-envelopes}) and outer ($r>$1600~au, see Sect. \\ref{sec:velocity-gradient-1000au}) parts of the envelopes.\n\n\\subsection{Characterization of rotational motions} \\label{sec:results-PV-diagram}\nWe assume that the protostellar envelopes are axisymmetric around their rotation axes, and thus, the velocity gradients observed along the equatorial axis, and reported in the PV$_\\mathrm{rot}$ diagrams, are mostly due to the rotational motions of the envelopes. We model the rotational velocity variations by a simple power-law model $\\mathrm{v} \\propto r^{\\alpha}$ without taking the upper limits into account. This method has been tested with an axisymmetric model of collapsing-rotating envelopes by \\cite{Yen13}. As long as rotation dominates the velocity field on the line of sight, which depends on the inclination and flattening of the envelope, \\cite{Yen13} obtain robust estimates of the rotation motions at work in the envelopes.\nFirst, we fix the power-law index at $\\alpha$=-1 to compare to what is theoretically expected for an infalling and rotating envelope from a progenitor core in solid-body rotation \\citep{Ulrich76, Cassen81, Terebey84, Basu98}. The reduced $\\chi^{2}$ values of fits by an orthogonal least-square model are reported in the second column of Table \\ref{table:chi2-fit-profil-rotation}.\nThen, we let the power-law index vary as a free parameter: the best power-law index and the reduced $\\chi^{2}$ found for each protostellar envelope in our sample are reported in the third column of Table \\ref{table:chi2-fit-profil-rotation}.\nFigure \\ref{fig:PV-diagrams-1} shows the PV$_\\mathrm{rot}$ diagrams adjusted by a power-law for the sources of the CALYPSO sample. \n\n\n\\begin{table}[h]\n\\centering\n\\caption{Parameters of best power-law fits to the PV$_\\mathrm{rot}$ diagrams.}\n\\label{table:chi2-fit-profil-rotation}\n\\resizebox{\\hsize}{!}{\\input{tables\/Table-fits-PVdiagram}}\n\\tablefoot{\n\\tablefoottext{a}{Range of radii over which the PV$_\\mathrm{rot}$ diagrams were built and the fits were performed.}\n\\tablefoottext{b}{Number of degrees of freedom we used for the modeling and reduced $\\chi^{2}$ value associated with the best fit with a power-law function $\\mathrm{v} \\propto r^{-1}$.}\n\\tablefoottext{c}{Number of degrees of freedom we used for the modeling, index of fit with a power-law function ($\\mathrm{v} \\propto r^{\\alpha}$) and the reduced $\\chi^{2}$ value associated with this best fit model.} \n}\n\\end{table}\n\n\n\nThe power-law indices of the PV$_\\mathrm{rot}$ diagrams from our sample are between -1.1 and 0.8.\nFive sources (L1448-2A, IRAS2A, SVS13-B, L1527, and GF9-2) show rotational velocity variations in the envelope scaling as a power law with an index close to -1. This is consistent with the expected index for collapsing and rotating protostellar envelopes. The reduced $\\chi^2$ are $\\sim$1.5 for these sources except for IRAS2A and SVS13-B for which it is better ($\\sim$0.2).\nL1521F and L1157 show a power-law index close to 0 with a very low reduced $\\chi^2$ ($\\leq$0.3, see Table \\ref{table:chi2-fit-profil-rotation}). These flat PV$_\\mathrm{rot}$ diagrams ($\\mathrm{v}_\\mathrm{rot} \\sim$ constant) suggest differential rotation of the envelope with an angular velocity of $\\Omega=\\frac{v_\\mathrm{rot}}{r} \\propto r^{-1}$.\nFor two other sources (IRAS4B and IRAM04191), the best indices are compatible with -0.5, which could suggest Keplerian rotation at scales of $r<$1000~au. However, the reduced $\\chi^2$ are also satisfactory ($\\sim$1) when we fix the power-law index at $\\alpha$=-1 (see Table \\ref{table:chi2-fit-profil-rotation}). Thus, for these two sources, our CALYPSO datasets only allow us to estimate a range of the power-law indices between -1 and -0.5 (see Table \\ref{table:chi2-fit-profil-rotation}).\n\nRotational velocity variations along the equatorial axis between 50 and 5000~au in L1448-NB cannot be reproduced satisfactorily by any single power-law model ($\\chi^2 >$2, see Table~\\ref{table:chi2-fit-profil-rotation}).\nHowever, considering only the points at $r<$400~au for L1448-NB, we obtain a power-law index of -0.9$\\pm$0.2 with a good reduced $\\chi^2$ of 0.4, as expected for a collapsing and rotating envelope (see Table \\ref{table:chi2-fit-profil-rotation}).\n\nWe found a positive index $\\alpha$ for IRAS4A of 0.8 (see Table \\ref{table:chi2-fit-profil-rotation}). It could be an indication of solid-body rotation of $\\Omega=\\frac{v_\\mathrm{rot}}{r} \\sim$constant. However, we observe that the velocity in the PV$_\\mathrm{rot}$ diagram decreases from 2000 to $\\sim$600~au and re-increases at small scales (see panel (f) of Fig. \\ref{fig:PV-diagrams-1}). Thus, the velocity gradient is not uniform on the scales traced by the PV$_\\mathrm{rot}$ diagram as would be expected for a solid-body rotation ($\\mathrm{v}_\\mathrm{rot} \\propto r$). \nMoreover, points at radii $<$600~au are consistent with an infalling and rotating envelope (see panel (f) of Fig. \\ref{fig:PV-diagrams-1}): considering only these points, we obtain a power-law index of -1.3$\\pm$0.6 with a good reduced $\\chi^2$ value of 0.6 (see Table \\ref{table:chi2-fit-profil-rotation}). There is a dip in the C$^{18}$O emission at $r<$350~au that could be due to the opacity (see Figures \\ref{fig:intensity-maps-IRAS4A} and \\ref{fig:column-density-maps-IRAS4A}), thus, below this radius the information on velocities could be altered. \nTo date, no observations have identified any solid-body rotating protostellar envelope. Numerical models also favor differential rotation of the envelope \\citep{Basu98}. The interpretation of the velocity field as tracing solid-body rotation in the envelope of IRAS4A is therefore unlikely to be correct.\n\nFor the sources IRAS2A, IRAM04191, and L1157, the reduced $\\chi^2$ is also good ($\\sim$1) when the PV$_\\mathrm{rot}$ diagrams of these sources are ajusted by a model with a fixed index of $\\alpha$=-1 (see Table \\ref{table:chi2-fit-profil-rotation}). We determine position and velocity from four different and independent methods and we did not consider the uncertainties of the connection between the different tracers and datasets. The uncertainty on the indices reported in Table~\\ref{table:chi2-fit-profil-rotation} may thus be underestimated.\nOn the other hand, although we determined the systemic velocity by maximizing the overlap of the blue and red points, this method does not allow a more accurate determination than 0.05~km~s$^{-1}$. The systematic error of 0.05~km~s$^{-1}$, added to previous velocity errors of the points in the PV$_\\mathrm{rot}$ diagrams to take this uncertainty on the systemic velocity into account (see Appendix \\ref{sec:systemic-velocity}), can be overestimated and thus lead us to underestimate the $\\chi^2$. For these three sources, the CALYPSO dataset only allow us to estimate a range of power-law indices between -1 and the $\\alpha$ value reported in the fifth column of Table~\\ref{table:chi2-fit-profil-rotation}.\nThe uncertainties on the indices reported in Table \\ref{sec:systemic-velocity} are statistical errors and a systematic uncertainty of $\\pm$0.1 has to be added to account for the uncertainties in the outflow directions and thus the equatorial axis directions (see Table \\ref{table:sample}).\nMoreover, despite the choice of the equatorial axis, the rotational velocities could be contaminated by infall at the small scales along this axis due to the envelope geometry.\n\nTo conclude, the organized motions reported in the PV$_\\mathrm{rot}$ diagrams and modeled by a power-law function with an index $\\alpha$ ranging from -2 to 0 are consistent with differential rotational motions ($\\Omega \\propto r^{\\epsilon}$, with -3$< \\epsilon <$-1 here). We identified rotational motions in all protostellar envelopes in our sample except in IRAS4A.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Distribution of specific angular momentum in the CALYPSO Class~0 envelopes} \\label{subsec:distribution-j}\n\n\\subsubsection{Specific angular momentum due to rotation motions}\n\nAssuming the motions detected along the equatorial axis are dominated by differential rotation for 11 of the 12 sources in our sample, we use the measurements reported in the PV$_\\mathrm{rot}$ diagrams to derive the radial distribution of specific angular momentum in the protostellar envelopes due to rotation. In this part of the study, IRAS4A is excluded. The specific angular momentum is $j=\\frac{J}{M}=\\frac{I \\Omega}{M}$ with the moment of inertia $I$ defined as $I \\propto M r^2$ \\citep{Belloche13}. Thus, the specific angular momentum is calculated from the rotational velocity: $j= \\mathrm{v}_\\mathrm{rot}(r) \\times r$. \nWe plot all the specific angular momentum profiles obtained for the CALYPSO subsample in panel (b) of Fig. \\ref{fig:diagramme-j-Belloche13+CALYPSO}.\nThe individual distribution of specific angular momentum $j(r)$ for each source is given in Appendix \\ref{sec:comments-indiv-sources}. This is the first time that the specific angular momentum distribution as a function of radius within a protostellar envelope is determined homogeneously for a large sample of 11 Class~0 protostars.\nWe performed a least-square fit of the $j(r)$ profiles for each source individually, using a model of a simple power-law and a broken power-law model to identify the change of regimes. \nThe broken power-law model function is defined with a break radius $r_\\mathrm{break}$ as follows:\n$$j(r) \\propto \\left( \\frac{r}{r_\\mathrm{break}} \\right)^{\\beta_{1}}~~\\mathrm{when}~~r<r_\\mathrm{break},$$\n$$j(r) \\propto \\left( \\frac{r}{r_\\mathrm{break}} \\right)^{\\beta_2}~~\\mathrm{when}~~r>r_\\mathrm{break}.$$\nWe report in Table \\ref{table:chi2-fit-profil-moment-ang} the power-law indices fitting the best individual profiles and the associated reduced $\\chi^2$. For the broken power-law fits, only results with a reduced $\\chi^2$ better than the one obtained with a simple power-law model and with a break radius value $r_\\mathrm{break}$ to which the $j(r)$ profile is really sensitive, have been retained.\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[scale=0.7,angle=0,trim=0cm 4cm 0cm 5cm,clip=true]{figures-discussion\/plot-diagramme-j-Belloche13+CALYPSO.pdf}\n\\caption{Radial distribution of specific angular momentum. Panel (a): Figure adapted from Figure 8 of \\cite{Belloche13} and from \\cite{Ohashi97}. Panel (b): zoom on the region where the angular momentum profiles due to rotation of the CALYPSO sources lie. The gray curve shows the median profile $j(r)$. In the two panels, the solid black line shows the best fit with a broken power-law model.\n}\n\\label{fig:diagramme-j-Belloche13+CALYPSO}\n\\end{figure*}\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=10cm]{figures-discussion\/comparaison-profil-specific-angular-momentum-log-tous-gradients.pdf}\n\\caption{Radial distribution of apparent specific angular momentum $|j_\\mathrm{app}|= |v| \\times r$ along the equatorial axis of the CALYPSO sources, considering all the velocity gradients observed at all envelope scales, including the reversed gradients and the shifted ones at scales of $r \\gtrsim$1600~au (see Fig. \\ref{fig:evolution-theta-plot} and Sect. \\ref{sec:velocity-gradient-1000au}) which were excluded in the construction of the PV$_\\mathrm{rot}$ diagrams in Fig. \\ref{fig:PV-diagrams-1}, and in panel (b) of Fig. \\ref{fig:diagramme-j-Belloche13+CALYPSO} for our analysis of rotational motions. The empty circles show the negative apparent specific angular momentum from the reversed gradients along the equatorial axis in the outer scales ($r >$1600~au) of the L1448-2A, IRAS2A, IRAM04191, L1527, L1157 and GF9-2 envelopes (see Sect. \\ref{subsec:Counter-rotation}). The gray curve shows the median profile |$j_\\mathrm{app}$| and the solid black line shows the best fit with a broken power-law model. }\n\\label{fig:profil-j-CALYPSO-tous-gradients}\n\\end{figure*}\n\nTwo sources (L1448-NB and L1448-C) are better reproduced by a broken power-law model than a simple power-law model where the $\\chi^2$ are $\\sim$2: this allows us to identify a change of slope from a relatively flat profile to an increasing profile at larger radius in the envelope ($\\beta \\sim$1), with break radii between 500 and 700~au. \nFor the other sources, we also identified at scales of $r<$1300~au a flat profile of specific angular momentum with $\\beta<0.5$ (L1448-2A, IRAS2A, SVS13-B, IRAS4B, L1527, and GF9-2) while the specific angular momentum profile at scales of $r>$1000~au shows a steeper slope with $\\beta \\sim$1 (L1521-F). \nHowever, two sources of the sample (IRAM04191 and L1157) stand out as the sources showing a steep increase in their specific angular momentum profile at scales of $r<$1000~au ($\\beta \\geq$0.7), similar to the indices found at large radii in the sources showing a break in their $j(r)$ profiles.\nWe note that for the flat profiles ($\\beta <$0.5; L1448-2A, SVS13-B, IRAS4B, and GF9-2), and IRAM04191 and L1157, the specific angular momentum distribution is only constrained at scales $<$1300~au.\nMost of the sources in our sample are better reproduced by a broken power-law model with a break radius (1000 $\\pm$ 500)~au and an increasing profile at larger radius in the envelope ($\\beta \\sim$1.4) than a simple power-law model.\n\n\nIn his review, \\cite{Belloche13} plotted the observed specific angular momentum as a function of rotation radius for several objects along the star-forming sequence. In this plot (panel (a) of Figure \\ref{fig:diagramme-j-Belloche13+CALYPSO}), he identifies three regimes in the distribution of specific angular momentum, that can be broadly associated with different evolutionary stages: \\\\\n\\indent $\\bullet$ prestellar regime: on large scales, the apparent angular momentum of molecular clouds \\citep{Goldsmith85} and dense cores \\citep{Goodman93, Caselli02} appears to follow the power-law relation $j \\propto r^{1.6}$,\\\\\n\\indent $\\bullet$ protostellar regime: between 100~au and $\\sim$6000~au (0.03~pc), a few points in different protostellar envelopes suggest the specific angular momentum is relatively constant ($j \\sim 10^{-3}$~km~s$^{-1}$~pc, \\citealt{Ohashi97, Belloche02, Chen07}),\\\\\n\\indent $\\bullet$ disk and binary regime: below 100~au, measurements in disks and Class~II binaries \\citep{Chen07} show a decrease of $j$ following a trend characteristic of Keplerian rotation ($j \\propto r^{0.5}$).\\\\\n\nThus, from previous observational studies on rotational motions, finding a break at $r \\sim$1000~au between two trends of specific angular momentum within Class~0 protostars was unexpected. Although the velocity gradients observed in the outer part of the protostellar envelopes ($r>$1000~au) are consistent with rotational motions, the observed $j$ regime at these scales is not expected from pure rotational motions.\n\n\n\n\\subsubsection{Apparent specific angular momentum}\n\nThe radius range of $j(r)$ distribution due to rotation motions is not homogeneous between sources (see Table \\ref{table:chi2-fit-profil-moment-ang}). To identify whether the radius of $\\sim$1000~au is a critical radius between two trends of specific angular momentum in each source, we derive the radial distribution of the apparent specific angular momentum |$j_\\mathrm{app}$| at all scales in the envelopes. To build |$j_\\mathrm{app}$|$(r)$ distribution, we consider the gradients observed at all envelope scales, including also the reversed gradients and the shifted ones at scales of $r \\gtrsim$1000~au (see Fig. \\ref{fig:evolution-theta-plot} and Sect. \\ref{sec:velocity-gradient-1000au}) which were excluded in the construction of the PV$_\\mathrm{rot}$ diagrams in Fig. \\ref{fig:PV-diagrams-1} because they are not consistent with rotational motions. By considering these velocity gradients, we add points in the outer envelopes but the trend observed in the inner envelopes do not change (see Tables \\ref{table:chi2-fit-profil-moment-ang} and \\ref{table:chi2-fit-profil-japp}). Thus, the |$j_\\mathrm{app}$|$(r)$ distribution helps us to understand the origin of the trend and the velocity gradients observed at $r >$1000~au.\nWe plot all the apparent specific angular momentum profiles obtained for the CALYPSO subsample in Fig. \\ref{fig:profil-j-CALYPSO-tous-gradients}. We also report the apparent specific angular momentum of IRAS4A which was identified as the only source that did not show any rotational motions in our sample (see Sect. \\ref{sec:results-PV-diagram}). As for $j$ profiles, we performed a least-square fit of the |$j_\\mathrm{app}$|$(r)$ profiles for each source individually and we report the indices of the power-law models in Table \\ref{table:chi2-fit-profil-japp}.\n\n\nWe create the median |$j_\\mathrm{app}$|$(r)$ profile of the CALYPSO subsample. We first resampled the individual profile of each source in steps of 100~au and normalized it by the value at 600~au, then we took the median value of individual profiles at each radius step. The median profile is shown in gray on Figure \\ref{fig:profil-j-CALYPSO-tous-gradients}.\nFrom a broken power-law fit, we obtain a relatively flat profile ($j_\\mathrm{app} \\propto r^{0.3 \\pm 0.3}$) at radii smaller than 1570$\\pm$300~au and an increasing profile ($j_\\mathrm{app} \\propto r^{1.6 \\pm 0.2}$) in the outer envelope. The radius of $\\sim$1600~au therefore appears to be a critical radius which delimits two regimes of angular momentum in protostellar envelopes: the specific angular momentum decreases down to $\\sim$1600~au and then tends to become constant.\n\n\nThe change of behavior of $j_\\mathrm{app}$ above the break radius could be due to a change of tracer to study the kinematics in the outer envelope. However, we do not find any systematic consistency between $r_\\mathrm{app,break}$ and the transition radius $R_\\mathrm{trans}$ between the two tracers C$^{18}$O and N$_{2}$H$^{+}$. Even if for SVS13-B, $R_\\mathrm{trans}$ is in the error bars of $r_\\mathrm{app,break}$, for three sources (L1448-NB, L1448-C, and IRAS4A) it is not consistent, and for IRAS4B, we do not observe a change of regime for $j_\\mathrm{app}$ at $r \\sim$1600~au (see Tables \\ref{table:chi2-fit-profil-moment-ang} and \\ref{table:chi2-fit-profil-japp}). Moreover, for L1521F, only the C$^{18}$O emission shows a velocity gradient allowing us to constrain the kinematics at scales of $r>$1600~au (see Fig. \\ref{fig:velocity-maps-L1521F}) and we find the same trend of $j_{app}$ ($\\beta_\\mathrm{app} \\sim$1.2) than in all other sources where we used N$_{2}$H$^{+}$ to constrain the outer part of the envelopes. \nThe other sources (L1448-2A, IRAS2A, IRAM04191, L1527, L1157, and GF9-2) show a negative value of the apparent angular momentum at outer envelope scales due to a renversal of the velocity gradients (see Fig. \\ref{fig:profil-j-CALYPSO-tous-gradients}, Table \\ref{table:chi2-fit-profil-japp}, and Sect. \\ref{subsec:Counter-rotation}). For two of these sources (L1448-2A and GF9-2) the radius where the gradient reverses along the equatorial axis, resulting in a negative $j_\\mathrm{app}$ with respect to the inner envelope scales, is consistent with $R_\\mathrm{trans}$ and $r_\\mathrm{app,break}$. For two sources (IRAS2A and IRAM04191), $R_\\mathrm{trans}$ is consistent with the radius where the gradient reverses along the equatorial axis but not with $r_\\mathrm{app,break}$. For the last two sources (L1527 and L1157), the three radii are all different from each other.\nThe different individual behaviors in the CALYPSO sample allow us to conclude that our finding that protostellar envelopes are characterized by two regimes of angular momentum does not result from our use of two different tracers.\n\n\nFrom the median |$j_\\mathrm{app}$|$(r)$ profile without normalization of the individual profiles at 600~au, we find a mean value of specific angular momentum in the inner parts of the envelopes ($r<$1600~au) of $\\sim$6 $\\times$10$^{-4}$~km~s$^{-1}$~pc. This value is slightly lower but compatible with the estimates made by \\cite{Ohashi97} and \\cite{Chen07} in four Class~0 or I sources ($j \\sim$10$^{-3}$~km~s$^{-1}$~pc at $r<$5000~au). It is also consistent with the studies by \\cite{Yen15b} and \\cite{Yen15} which find values between 5 $\\times$ 10$^{-3}$~km~s$^{-1}$~pc and 5 $\\times$ 10$^{-5}$~km~s$^{-1}$~pc in the inner envelope (r$<$1500~au). \n\\cite{Yen15b} estimate a specific angular momentum of $\\sim 5 \\times$ 10$^{-4}$~km~s$^{-1}$~pc at $r \\sim$100~au for L1448-C and L1527. Moreover, our values for L1157 are consistent with their upper limit estimate of $5 \\times$ 10$^{-5}$~km~s$^{-1}$~pc in the inner envelope ($r<$100~au) of L1157. \n\n\nThe high angular resolution and the high dynamic range of the CALYPSO observations allow us to identify the first two regimes within individual protostellar envelopes: values at radii $\\gtrsim$1600~au ($j_\\mathrm{app} \\propto r^{1.6}$ on average, see Table \\ref{table:chi2-fit-profil-japp}) seem to correspond to the trend found in dense cores at scales $>$6000~au while the values stabilize around $\\sim$6 $\\times$10$^{-4}$~km~s$^{-1}$~pc on average at radii $<$1600~au. This study resolves for the first time the break radius between these two regimes deeper within the protostellar envelopes at around $\\sim$1600~au instead of $\\sim$6000~au. \nIn a study of ammonia emission in the outer envelopes of two Class~0 objects, \\cite{Pineda19} find an increasing angular momentum profile scaling as $r^{1.8}$ from 1000~au to 10000~au, with values $\\sim$3 $\\times$10$^{-4}$~km~s$^{-1}$~pc at radii $\\sim$1000~au. They do not detect the break around $\\sim$1600~au found in the CALYPSO sample.\nThis break radius from which the profiles are found to be flat in the inner envelope may depend on the evolutionary stage of the accretion process during the Class~0 phase as suggested by \\cite{Yen15}. It could be due to the propagation of the inside-out expansion wave during the collapse \\citep{Shu77}: assuming a median lifetime or half life of $\\sim$5 $\\times$10$^4$~yr for Class~0 protostar envelopes (\\citealt{Maury11}; see also \\citealt{Evans09}) at sound velocity ($\\sim$ 0.2~km~s$^{-1}$), one obtains a radius $\\sim$2000~au.\nThis radius is on the same order of magnitude as the observed break radius between the two regimes observed in the distribution of specific angular momentum of sources in our sample. In this case, the break radius could be an indication of the age of the protostars, except for four sources (L1448-NB, IRAM04191, L1521F, and L1157) in our sample where we do not observe this break radius. \nBeyond this radius, the outer envelope may not have collapsed yet, and could therefore retain the initial conditions in angular momentum of the progenitor prestellar core. \n\nThis could be an explanation for the increase in angular momentum observed at the scales of $r>$1600~au ($j_\\mathrm{app} \\propto r^{1.6}$ on average, see Table \\ref{table:chi2-fit-profil-japp}), consistent with the prestellar stage ($j \\propto r^{1.6}$). \nWe discuss the properties and physical origin of these two regimes in more details in the next sections.\n\n\n\n\n\n\\subsection{Conservation of angular momentum in Class~0 inner envelopes} \\label{sec:j-inner-envelopes}\n\n\nIn this section, we focus on the relatively constant values of specific angular momentum observed in the inner envelopes at scales of $r \\le$1600~au in the $j(r)$ profiles due to rotation motions (see Fig. \\ref{fig:diagramme-j-Belloche13+CALYPSO}). From these flat profiles, we find that the matter directly involved in the formation of the stellar embryo has a specific angular momentum $\\sim$3 orders of magnitude higher than the one in T-Tauri stars ($j \\sim$2 $\\times$10$^{-7}$~km~s$^{-1}$~pc, \\citealt{Bouvier93}). We discuss constant values of specific angular momentum as conservation of angular momentum to test disk formation as a possible solution to the angular momentum problem.\n\n\nIt is difficult to constrain the time evolution of specific angular momentum for a given particle from angular momentum distributions which are snapshots of the angular momentum distribution of all particles at a given time during the collapse phase. During the collapse of a core initially in either solid-body rotation or differential rotation, particles conserve their specific angular momentum during the accretion on the stellar embryo \\citep{Cassen81,Terebey84, Goodwin04}. \nIn the case of a protostellar envelope with a density profile $\\rho \\propto r^{-2}$, an observed flat profile $j(r)=$constant requires, since each particle at different radii has the same specific angular momentum, an initially uniform distribution of angular momentum. This does not agree with the steep increase in specific angular momentum we observe at scales of $r>$1600~au in the $j(r)$ profiles. The break in the specific angular momentum profile could be due either to a faster collapse of the inner envelope caused by an initial inner density plateau \\citep{Takahashi16} or to a change of dominant mechanisms responsible for the observed velocity gradients from inner to outer scales of the envelope.\n\n\nIn our sample, we distinguish eight sources with a relatively flat $j(r)$ profile in the inner envelope ($\\beta<$0.5, see Table \\ref{table:chi2-fit-profil-moment-ang}): L1448-2A, L1448-NB, L1448-C, IRAS2A, SVS13-B, IRAS4B, L1527, and GF9-2. \nWe estimate a centrifugal radius that would be obtained when the mass currently observed at $\\sim$100~au collapses and based on the mean value of specific angular momentum observed today $<j_\\mathrm{100~au}>$ as follows:\n\\begin{equation}\nR_\\mathrm{cent}=\\frac{<j_\\mathrm{100~au}>^2}{G~M_\\mathrm{100~au}}.\n\\label{eq:Rcent}\n\\end{equation}\n\n\nThe lower limit of the mass enclosed within 100~au, $M_\\mathrm{100~au}$, is the mass of the envelope $M_\\mathrm{100~au}^\\mathrm{dust}$ estimated from the PdBI 1.3~mm dust continuum flux \\citep{Maury18}, assuming optically thin emission, a dust temperature at 100~au computed with Eq. \\eqref{Tdust} and corrected by the assumed distance (see Table \\ref{table:sample}). This mass estimate does not include the mass of the central stellar object, $M_{\\star}$: since the embryo mass is unknown for most sources in our sample, we consider an upper limit of $M_\\mathrm{100~au}=M_{\\star}+M_\\mathrm{100~au}^\\mathrm{dust}$ assuming $M_{\\star}=$ 0.2~M$_{\\odot}$ for each source in our sample. This value of 0.2~M$_{\\odot}$ corresponds to the stellar mass in the Class~0\/I protostar L1527 from kinematic models of the Keplerian pattern in the disk \\citep{Tobin12, Ohashi14, Aso17}. The range of values for $M_\\mathrm{100~au}$ are reported in the third column in Table \\ref{table:rayon-disque-brot}. The calculated range of centrifugal radii associated with $M_\\mathrm{100~au}$ is listed for each source in the fourth column in Table \\ref{table:rayon-disque-brot}. We note that if $M_{\\star}$ of a source is smaller than that of L1527, then the centrifugal radius value we calculated is underestimated.\n\nSince the embryo mass is uncertain and $M_\\mathrm{100~au}$ may be underestimated if the dust emission is not optically thin, we compute the mass enclosed within $r<$100~au, including the stellar embryo mass, needed to form a disk the size of $R_\\mathrm{disk}^\\mathrm{dust}$ with the $<j_\\mathrm{100~au}>$ observed. The values are reported in the last column of Table \\ref{table:rayon-disque-brot}.\n\n\n\n\\begin{table*}[!ht]\n\\centering\n\\caption{Centrifugal radius $R_\\mathrm{cent}$ assuming angular momentum conservation.}\n\\label{table:rayon-disque-brot}\n\\input{tables\/Table-Rcent}\n\\tablefoot{\n\\tablefoottext{a}{Weighted mean of specific angular momentum in the inner envelopes (50~au$< r \\le$1600~au).}\n\\tablefoottext{b}{Range of the object mass at 100~au, the minimum and maximum values are defined in Sect. \\ref{sec:j-inner-envelopes}.}\n\\tablefoottext{c}{Centrifugal radii estimated from $<j_\\mathrm{100~au}>$ and $M_\\mathrm{100~au}$ using Eq. \\eqref{eq:Rcent}, assuming conservation of angular momentum.}\n\\tablefoottext{d}{Candidate disk radius determined from the CALYPSO study of PdBI dust continuum emission at 1.3 and 3~mm \\citep{Maury18}, corrected by the assumed distance (see Table \\ref{table:sample}).}\n\\tablefoottext{e}{Total minimum mass that needs to be enclosed at $r<$100~au to form a disk equal to $R_\\mathrm{disk}^\\mathrm{dust}$ if the angular momentum $<j_\\mathrm{100~au}>$ was conserved. This minimum mass considers the mass of the stellar embryo and the mass of the optically thick inner envelope enclosed within 100~au.}\n}\n\\end{table*}\n\n\nFor all the sources in our sample, the upper limits of the $R_\\mathrm{cent}$ range are larger than 150~au and systematically larger than the continuum disk candidate radii $R_\\mathrm{disk}^\\mathrm{dust}$ from \\cite{Maury18} reported in the fifth column of Table \\ref{table:rayon-disque-brot}. \nMoreover, \\cite{Maret20} only detect possible Keplerian rotation in two protostars in our sample (L1527 at radii $\\sim$90~au and L1448-C at $r \\sim$200~au) from the CALYPSO data. Thus, most $R_\\mathrm{cent}$ values is expected to be less than 100~au. The upper $R_\\mathrm{cent}$ values are probably overestimated because the contribution of the embryo mass to $M_\\mathrm{100~au}$ is excluded.\n\n\n\n\nComparing the lower limits of the $R_\\mathrm{cent}$ range with the candidate disk radius, we find a good agreement for most sources in our sample except for L1448-NB.\nWe find a larger centrifugal radius ($\\sim$500~au) than the observed disk size ($<$50~au) calculated considering only the main protostar L1448-NB1 of the binary system. Since in this study, we are interested in the kinematics of the whole system, we must consider all the continuum structure and not only that of the main protostar. Considering NB1 and NB2, \\cite{Maury18} resolve a circumbinary structure with a radius of (320 $\\pm$ 90)~au centered on the middle of the two components. Given the uncertainties, the latter value is consistent with the lower centrifugal radius estimated here. At these scales, \\cite{Tobin16} observe a spiral structure surrounding the multiple system and interpreted it as a gravitationally unstable circumbinary disk. On the other hand, \\cite{Maury18} suggest that this component is due to orbital motions and tidal arms between the companions and \\cite{Maret20} do not detect any Keplerian rotation at radii $<$170~au. Thus, the nature of this additional structure surrounding the multiple system is still unclear. As a consequence, the increase in specific angular momentum we measured at small scales could not only trace the rotation of {the disk or} the envelope but may be contaminated by gravitational instabilities due to orbital motions or a fragmented disk surrounding the system.\nGiven the large uncertainties on the dust disk radii, we found a good agreement between centrifugal radii and $R_\\mathrm{disk}^\\mathrm{dust}$ for L1448-C and L1527. Moreover, the dust radius (50~au in L1527, \\citealt{Maury18}) does not necessarily exactly correspond to the centrifugal radius which was first detected in L1527 from observations of SO emission at 100$\\pm$20~au \\citep{Sakai14Nat}. For this source, our estimate of $R_\\mathrm{cent}$ ($\\sim$70~au) is consistent with previous kinematic studies which detect a proto-planetary disk candidate with a radius of 50$-$90~au \\citep{Ohashi14, Aso17, Maret20}. Moreover, we observe a slight increase in the specific angular momentum we measured at $r<$80~au. It could be due to the transition from the envelope to the disk.\n\n\n\nThe hypothesis of collapsing material with conservation of angular momentum, resulting in disk formation, at $r<$100~au is therefore plausible for most sources in our sample. \nWe notice that L1448-NB, in which \\cite{Tobin16} claim the detection of a large candidate disk, shows the highest value of specific angular momentum at $r<$1600~au of the CALYPSO sample, consistent with the angular momentum observed in the proto-planetary disks surrounding the T-Tauri stars which are estimated to be 1$-$6$\\times$10$^{-3}$~km~s$^{-1}$~pc \\citep{Simon00,Kurtovic18,Perez18}. It could suggest an increase in the angular momentum of the disk during its evolution. In this case, the mean value of $j(r)$ in the inner envelope would be lower in the less evolved than in the more evolved Class~0 protostars, and it would increase with time until reaching the value contained in the T-Tauri disks. In this scenario, L1448-NB would be one of the most evolved objects in the sample. However, the borderline Class~0\/I protostar L1527, which is the most evolved object of the CALYPSO sample, has a specific angular momentum of $\\sim$6 $\\times$ 10$^{-4}$~km~s$^{-1}$~pc at the inner envelope scales (see Table \\ref{table:rayon-disque-brot}). In the same way, L1448-C has a specific angular momentum less than one order of magnitude lower than the values observed in the Class~II disks while \\cite{Maret20} suggest the presence of a Keplerian disk in the inner envelope. As most of the CALYPSO inner protostellar envelopes have an order of magnitude less specific angular momentum than in Class II disks, we discuss below several possible explanations:  \\\\\n(i) a part of the angular momentum inherited by the T-Tauri disks may not come from the rotating matter contained in the inner envelope accreted during the Class~0 phase. During the Class~I phase, the mass accreted could come from regions further away from the envelope ($r \\gg$1600~au) with a possibly higher specific angular momentum.\\\\\n(ii) disks may expand with time due to the transfer of angular momentum from their inner regions to their outer ones. Unfortunately, the specific angular momentum does not contain information about the mass. Large values of $j(r)$ may be carried by low masses at the outer disk radius but may remain difficult to quantify. To this day, the mechanisms at work in disk evolution remain an open question. Some studies, for example, suggest that viscous friction may be responsible for the disk expansion \\citep{Najita18}. \\\\\n(iii) the specific angular momentum of the proto-planetary disks may be biased toward high values from historical, large and massive disks. A new population of small T-Tauri disks with radii between 10 and 30~au has been observed thanks to PdBI and ALMA \\citep{Pietu14, Cieza19}. Assuming a small rotationally-supported disks around a stellar object including a total mass of 0.1$-$1~$M_{\\odot}$ \\citep{Pietu14}, one expects a specific angular momentum between 10$^{-5}$~km~s$^{-1}$~pc and 10$^{-4}$~km~s$^{-1}$~pc, values which are similar to those we obtained in the inner Class~0 protostellar envelopes with the CALYPSO sample. However, to this day, no resolved observations of gas kinemactics of these small Class II disks allow us to estimate observationally their specific angular momentum.\\\\\n\n\n\n\n\n\n\\subsection{Origin of the velocity gradients at $r>$1600~au} \\label{sec:velocity-gradient-1000au}\nAt outer envelope scales, we detect velocity gradients ($\\sim$2~km~s$^{-1}$~pc$^{-1}$ at $\\sim$10000~au, see Table \\ref{table:gradient-velocity-fit}) in the CALYPSO single-dish maps. They may not be directly related to rotational motions of the envelopes but rather to other mechanisms. Indeed, we observe in the CALYPSO dataset a systematic evolution of the orientation of the gradients between the inner and outer scales in the envelope (see Table \\ref{table:gradient-velocity-fit}). Figure \\ref{fig:evolution-theta-plot} shows the orientation of the mean velocity gradient observed at different scales of the envelope with respect to the position angle of the gradient observed at scales $\\sim$100~au. The clear dispersion ($\\sim$100$^{\\circ}$ on average, see Fig. \\ref{fig:evolution-theta-plot}) of gradient position angle across scales within individual objects may be due to a change of dominant mechanisms responsible for the observed gradients from inner to outer scales of the envelope. \nFrom the literature, velocity gradients are often measured in the outer protostellar envelopes along the equatorial axis and they are interpreted as due to rotational motions or infall from a filamentary structure at scales of 1500$-$10000~au \\citep{Ohashi97, Belloche02, Tobin11}. In this section, we explore the possible origins of the velocity gradients found at scales of $r>1600$~au {and used to build the |$j_\\mathrm{app}$|$(r)$ profiles (see Fig. \\ref{fig:profil-j-CALYPSO-tous-gradients}).}\n\n\n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.45,angle=0,trim=6.8cm 0cm 6.8cm 3cm,clip=true]{figures-discussion\/evolution-velocity-gradients.pdf}\n\\caption{Evolution of the orientation of the mean velocity gradient in the different datasets used to build the PV$_\\mathrm{rot}$ diagrams and angular momentum distributions with respect to the PA of the velocity gradient observed at small scales $\\Theta_\\mathrm{small}$ (PdBI C$^{18}$O emission). The error bars of the orientation $\\Theta$ are given in Table \\ref{table:gradient-velocity-fit}. They are smaller than $10^{\\circ}$ except for 7 of the 67 gradient measurements. For these 7 measurements, the large error bars are generally due to the absence of a clear gradient on either side of the central position of the source. Gradient measurements with large error are indicated by an empty circle. A typical error of $\\pm 10^{\\circ}$ is shown on the first point of the plot.}\n\\label{fig:evolution-theta-plot}\n\\end{figure}\n\n\n\n\\subsubsection{Questioning the interpretation of counter-rotation}\n\\label{subsec:Counter-rotation}\nSix sources in the sample show a clear reversal of the orientation of the mean velocity gradient ($| \\Theta - \\Theta_\\mathrm{small} | > 130^{\\circ}$) from the inner to the outer envelope scales: IRAS4A, IRAS4B, L1527, IRAM04191, L1157, and GF9-2. We note that the kinematics at scales where we observed reversed velocity gradients ($r>$1600~au) with respect to the small scales were not taken into account to build the PV$_\\mathrm{rot}$ diagrams in Fig. \\ref{fig:PV-diagrams-1}, or the specific angular momentum profiles shown for the full sample in panel (b) of Fig. \\ref{fig:diagramme-j-Belloche13+CALYPSO}. Indeed, these profiles were aimed at characterizing the rotational motions in the envelopes and the angular momentum due to this rotation: a reversal of the rotation, if real, would require a more complex model than the power-law ($\\mathrm{v} \\propto r^{\\alpha}$) model we adopted in Sects. \\ref{subsec:diagram-PV-construction} to \\ref{sec:j-inner-envelopes}. In this section, we discuss these complex patterns in more detail.\n\nIn IRAM04191, we observed velocity gradients at outer envelope scales of $r>$1600~au consistent with those observed previously by \\cite{Belloche02} and \\cite{Lee05} ($\\Theta \\sim$100$^{\\circ}$, see Table \\ref{table:gradient-velocity-fit}). However, in the inner envelope, we noticed a velocity gradient with a direction of $\\Theta =$-83$^{\\circ}$ (see bottom middle panel in Fig. \\ref{fig:velocity-maps-IRAM04191} and Table \\ref{table:gradient-velocity-fit}). In L1527, we found small-scale velocity gradients ($\\Theta \\sim$0$^{\\circ}$ at $r \\sim$ 1000~au) consistent with those previously observed by \\cite{Tobin11} which are in the opposite direction compared to the large-scale one ($r \\sim$8000~au, \\citealt{Goodman93}). \\cite{Tobin11} interpret this reversal of velocity gradients as counter-rotation but it also could be due to infalling motions that dominated the velocity field at the outer envelope scales \\citep{Harsono14}. \n\nOur study suggests that reversals of velocity gradients are common in Class~0 protostellar envelopes. However, the asymmetrical velocity gradients (for IRAS4B, GF9-2), the filamentary structures traced by the integrated intensity at scales of $r>$2000~au (for IRAS4A, IRAS4B, L1527, and GF9-2), and a strong external compression of the cloud hosting IRAS4A and IRAS4B \\citep{Belloche06} lead us to exclude the observed reversed gradients as counter-rotation of the envelope. \nMoreover, only MHD models with Hall effect succeed to form envelopes in counter-rotation. These models form a thin layer of counter-rotating envelopes at the outer radius of the disk ($r \\sim$50-200~au; \\citealt{Tsukamoto17}). This envelope layer is in counter-rotation compared to the formed disk and the protostellar envelope at $r>$200~au as a consequence of the Hall effect generated by the rotation of the disk which changes the angular momentum of the gas at the disk outer radius. Therefore, these models cannot explain the inversions of rotation in the different layers of the envelope at scales of $r>$3000~au as observed in our sample.\nHistorically, the gradients observed from single-dish mapping at $r>$3000~au have been used to quantify the amplitude for the angular momentum problem. However, incorrectly interpreted as pure rotational motions in the envelope, the resulting angular momentum measurements and the expected disk radii would be significantly overestimated. \n\nRecent studies on the angular momentum of the protostellar cores from hydrodynamical simulations of star formation are questioning the standard model of star formation from a collapsing core initially in solid-body rotation \\citep{Kuznetsova19, Verliat20}. They show that the angular momentum of synthetic protostellar cores is not directly related to the initial rotation of the synthetic cloud, and Keplerian disks can be formed from a simple non-uniform perturbation in the initial density distribution.\nIn this scenario, the angular momentum observed in inner protostellar envelopes and disks may not been inherited from larger-scale initial conditions but generated during the collapse itself.\n\n\n\\subsubsection{Contribution of infalling motions and core-forming motions}\n\nThe misalignments between the gradients observed in the envelopes at inner and outer envelope scales suggest a change of dominant mechanisms at $r>$1600~au. At large scales, infalling motions of the envelope can dominate rotational motions. In the hypothesis of a flattened infalling envelope, infall motions are expected to produce a velocity gradient projected in the plane of the sky that is oriented along the minor axis of the envelope, namely at the same position angle as the outflow. \nIn L1448-NB, SVS13-B and L1527, we detect velocity gradients aligned with the outflow axis at $r>$3000~au while at small scales the gradients are consistent with the equatorial axis (see Table \\ref{table:gradient-velocity-fit}). These three sources could be good candidates of the transition from collapse to rotation between large and small scales.\nThis scenario is also suggested in the study of \\cite{Ohashi97b}. They suggested that at outer envelope scales of $r \\sim$2000~au, the protostellar envelope L1527 is not rotationally supported ($\\mathrm{v}_\\mathrm{rot} \\sim$0.05~km~s$^{-1}$) but is dominated by the collapse ($\\mathrm{v}_\\mathrm{inf} \\sim$0.3~km~s$^{-1}$). \n\nCurrently, there are very few constraints on the infall velocities at scales of $r>$1600~au in the CALYPSO protostellar envelopes. \\cite{Belloche02} estimate an infall velocity of $\\mathrm{v}_\\mathrm{inf} \\sim$0.15~km~s$^{-1}$ at $r \\sim$1000~au from radiative transfer modeling of CS and C$^{34}$S emission in IRAM04191. In the dense core L1544, \\cite{Tafalla98} suggest also an infall velocity of $\\sim$0.1~km~s$^{-1}$ at scales $>$3000~au. The velocity offset, with respect to the systemic velocity assumed\nfor each source, found along the equatorial axis at $>$1000~au with CALYPSO is reported in Table \\ref{table:velocity-100-1000au}.\nFor most sources, we find typical velocity offsets $\\lesssim$0.3~km~s$^{-1}$ at scales of 1600~au (see Table \\ref{table:velocity-100-1000au}), consistent with infall velocities found in IRAM04191 and L1544, except for IRAS4A.\nIRAS4A harbors a velocity of $\\sim$0.5~km~s$^{-1}$ at $r \\sim$1000~au. This result is consistent with those of \\cite{Belloche06} at $\\sim$2000~au. They suggest that a fast collapse is triggered by an external compression from the cloud in which the source is embedded.\nThus, for all sources in our sample, the velocity gradient misalignment could be due to a change of mechanism dominating the velocities projected on the line of sight. This suggests either rotational velocities much smaller than infall velocities or a non axisymmetric geometry of the kinematics at outer envelope scales.\n\n\n\nMoreover, \\textit{Herschel} observations have shown that most solar-type prestellar cores and protostars form in filaments \\citep{Andre14}. Indeed, the column density maps of the Herschel Gould Belt Survey program\\footnote{See \\url{http:\/\/gouldbelt-herschel.cea.fr\/archives}} \\citep{Andre10} reveal that the CALYPSO protostars are embedded in or lie in the immediate vicinity of filamentary structures with $N_{H_2} >$10$^{21}$~cm$^{-2}$. Thus, the large-scale kinematics in protostellar envelopes could be contaminated or dominated by the kinematics of the filaments. \n\\cite{Kirk13} studied the velocity field of Serpens-South in the Aquila molecular cloud and showed a complex kinematics with longitudinal collapse along the main filament, radial contractions, and accretion streams from the cloud to the main filament. The longitudinal collapse of the filament could be responsible for the large-scale gradients observed in our protostellar envelopes as observed in the Serpens-Main region by \\cite{Dhabal18}. \nSeveral studies also highlighted transverse velocity gradients perpendicular to the main filament that suggested the material may be accreting along perpendicular striations \\citep{Palmeirim13,Dhabal18,Arzoumanian18, Shimajiri19}.\n\\cite{Palmeirim13} estimate the velocity of the infalling material to be ~0.5$-$1~km~s$^{-1}$ at $r \\sim$0.4~pc in the B211\/L1495 region. In our velocity maps at 10000~au along the equatorial axis, we measure typical velocities $<$0.3~km~s$^{-1}$ in most of the sources (see Table \\ref{table:velocity-100-1000au}) except in L1448-2A, IRAS4A, and IRAS4B. These three sources exhibit velocities of 0.5$-$1~km~s$^{-1}$ consistent with infall velocities estimated at filamentary scales. In these cases, host-filament motions could dominate the kinematics in outer protostellar envelopes ($r>$1600~au). \n\n\n\n\n\\begin{table}[!ht]\n\\centering\n\\caption{Value of velocity offset, with respect to the systemic velocity assumed for each source, at 100, 1000, and 10000~au along the equatorial axis from the velocity maps and considering all the gradients observed, even those not consistent with rotational motions.}\n\\label{table:velocity-100-1000au}\n\\input{tables\/Table-velocity-100-1000au}\n\\end{table}\n\n\n\n\n\n\\subsubsection{Contamination by turbulent motions from cloud scales}\n\nAll sources of the CALYPSO subsample (except L1448-NB) reveal a steep increase in apparent specific angular momentum with the radius at $\\sim$1600$-$10000~au scales, with an average trend of $j_\\mathrm{app} \\propto r^{1.6 \\pm 0.2}$ (see Table \\ref{table:chi2-fit-profil-japp} and Fig. \\ref{fig:profil-j-CALYPSO-tous-gradients}). This trend is similar to that observed in prestellar cores and clumps at scales $>$10000~au (see Figs. \\ref{fig:diagramme-j-Belloche13+CALYPSO} and \\ref{fig:profil-j-CALYPSO-tous-gradients}). Indeed, \\cite{Goodman93}, \\cite{Caselli02} and \\cite{Tatematsu16} show a trend between the size of prestellar cores and their observed mean angular momenta at scales on the order of 0.1~pc: $j(r)$ distribution scaling as $r^{1.2-1.7}$. From this dependency of $j$ with core radius and the linewidth-size relation, \\cite{Tatematsu16} suggest that non-thermal motions (turbulence) are related to the origin of angular momentum observed in these 0.1~pc cores.\n\n\n\\cite{Burkert00} studied numerical models of turbulent molecular clouds with a symmetric density profile and a Gaussian or random velocity field. They showed that 0.1~pc cores with random motions exhibit most of the time velocity gradients that, interpreted as rotation, would have specific angular momentum values of $j \\sim 3 \\times 10^{-3}$~km~s$^{-1}$~pc (10$^{21}$~cm$^2$~s$^{-1}$) and would scale as $j \\propto r^{1.5}$. This is in good agreement with observed values at 0.1~pc from the literature \\citep{Goodman93,Caselli02,Tatematsu16}. Our observations showing a trend of $j \\propto r^{1.6}$ at scales $\\sim$1600-5000~au could be either a signature of the turbulent cascade from the large-scale ISM propagating with subsonic properties to 1600~au envelope scales, or gravitationally-driven turbulence due to large-scale collapse motions at the interface between filaments and cores \\citep{Kirk13}.\n\n\n\nFrom the analysis of the gas velocity dispersion in molecular line observations, \\cite{Goodman98} and \\cite{Pineda10} identify the dense cores at a typical scale of 0.1~pc as the first velocity-coherent structures decoupled from the turbulent cloud. In this case, we would expect a quiescent structure with subsonic motions at radii $<$0.1~pc and the interpretation of ISM turbulent cascade with supersonic motions as a consequence of the steep increase of $j_\\mathrm{app}$ at scales $<10000$~au would no longer be valid.\nExcept L1448-2A, IRAS4A, and IRAS4B which exhibit velocities 0.5$-$1~km~s$^{-1}$ consistent with supersonic turbulent motions, all sources in our sample show typical velocities $\\lesssim$0.3~km~s$^{-1}$ consistent with subsonic-transonic turbulent motions.\nThis could suggest that the power-law behavior of $j_\\mathrm{app} \\propto r^{1.6}$ observed in the outer envelopes ($r>$1600~au) could be a scaling law due to the tail of a low velocity subsonic-transonic turbulent cascade.\n\n\nAt scales of $r>$1600~au, we observe typical velocity linewidths $\\lesssim$1~km~s$^{-1}$ (see Fig. \\ref{fig:line-width-diagrams}). We note that the linewidths tend to decrease from $\\sim$1600~au to larger scales in the outer envelopes and they do not show scaling laws with the radius as expected from turbulent motions in the ISM (\\citealt{Larson81}; see Fig. \\ref{fig:line-width-diagrams} and Table \\ref{table:chi2-fit-profil-dV}), but the velocity structure at these scales seems to show multiple components in velocity for several sources (L1527, L1448-C, IRAS2A, SVS13-B, IRAS4A, IRAS4B). As we can not disentangle them and identify exactly which component comes from the outer envelope or the host cloud for example, we need either a more elaborate model than a Gaussian or a HFS model to analyze the spectra or a more suitable tracer to determine more robustly the linewidths of the outer envelopes.\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=9cm]{fig-L14482A\/plot-distribution-dV-L1448-2A.pdf}\n\\includegraphics[width=9cm]{fig-L1448N\/plot-distribution-dV-L1448-NB2.pdf}\n\\includegraphics[width=9cm]{fig-L1448C\/plot-distribution-dV-L1448C.pdf}\n\\includegraphics[width=9cm]{fig-IRAS2A\/plot-distribution-dV-IRAS2A.pdf}\n\\includegraphics[width=9cm]{fig-SVS13B\/plot-distribution-dV-SVS13B.pdf}\n\\includegraphics[width=9cm]{fig-IRAS4A\/plot-distribution-dV-IRAS4A.pdf}\n\\caption{Linewidth along the equatorial axis of the CALYPSO protostellar envelopes. Blue and red dots show the linewidths at positions that have blue- and red-shifted velocities, respectively. Dots and large dots show the C$^{18}$O and N$_{2}$H$^{+}$ data, respectively. The dashed curve shows the best fit with a power-law model leaving the index $\\gamma$ as a free parameter ($D \\mathrm{v} \\propto r^{\\gamma}$) in the outer envelope (see Appendix \\ref{sec:details-Dv-distribution}). The radius of the outer envelope is given by the break radius of the $j(r)$ or $j_\\mathrm{app}(r)$ profiles (see Tables \\ref{table:chi2-fit-profil-moment-ang} and \\ref{table:chi2-fit-profil-japp}) or the radius where we observe a reversal of the velocity gradients with respect to the inner envelope. The vertical dashed lines show the transition radii between the different datasets (PdBI, combined, and 30m) and the two tracers as given in Table \\ref{table:radius-lines}.\n}\n\\label{fig:line-width-diagrams}\n\\end{figure*}\n\n\n\n\\begin{figure*}[!ht]\n\\addtocounter{figure}{-1}\n\\centering\n\\includegraphics[width=9cm]{fig-IRAS4B\/plot-distribution-dV-IRAS4B.pdf}\n\\includegraphics[width=9cm]{fig-IRAM04191\/plot-distribution-dV-IRAM04191.pdf}\n\\includegraphics[width=9cm]{fig-L1521F\/plot-distribution-dV-L1521F.pdf}\n\\includegraphics[width=9cm]{fig-L1527\/plot-distribution-dV-L1527.pdf}\n\\includegraphics[width=9cm]{fig-L1157\/plot-distribution-dV-L1157.pdf}\n\\includegraphics[width=9cm]{fig-GF92\/plot-distribution-dV-GF92.pdf}\n\\caption{Continued.\n}\n\\end{figure*}\n\n\n\n\n\n\n\n\\section{Conclusions}\nIn the framework of the CALYPSO survey, we analyzed the kinematics of Class~0 protostellar envelopes. The main results of our study are listed below.\n\\begin{enumerate}\n\\item We identify differential rotation motions in 11 sources in a sample of 12 Class~0 protostellar envelopes. The only exception is IRAS4A : the motions reported in the PV$_\\mathrm{rot}$ and modeled by a power-law function are consistent with a solid-body rotation, but the velocity gradient is not uniform in the inner envelope at $r<$2000~au as would be expected.\n\n\\item This is the first time that the specific angular momentum distribution as a function of envelope radius is determined homogeneously for a large sample of 11 Class~0 protostars. The high angular resolution and the high dynamic range of the CALYPSO observations allow us to identify two distinct regimes: the apparent specific angular momentum decreases as $j_\\mathrm{app} \\propto r^{1.6 \\pm 0.2}$ down to $\\sim$1600~au and then tends to become relatively constant around $\\sim$6 $\\times$ 10$^{-4}$~km~s$^{-1}$~pc down to $\\sim$50~au. \n\n\\item The values of specific angular momentum measured in the inner Class~0 envelopes suggest that material directly involved in the star formation process ($<$1600~au) typically encloses the same order of magnitude in specific angular momentum as what is inferred in small T-Tauri disks ($r \\sim$10~au). The constant values of $j$ at 50$-$1600~au allow us to determine good estimates of the centrifugal radius in the Class~0 protostars of the CALYPSO sample, which compare well with the disk radii estimated from the dust continuum \\citep{Maury18}. This suggests that the specific angular momentum is conserved during the accretion on the stellar embryo, resulting in disk formation. \n\n\n\\item At scales of $r>$1600~au, we conclude that the velocity gradients observed in the outer envelope with respect to small scales are not due to pure rotational motions or counter-rotation motions but related to other mechanisms. Historically, the gradients observed from single-dish mapping at $r>$3000~au have been interpreted as rotation and used to quantify the amplitude of the angular momentum problem for star formation. Thus, if the gradients are incorrectly interpreted as rotation, the angular momentum problem for star formation and the expected disk radii may have been significantly over-estimated. Moreover, we find no robust hints that envelopes are rotating with typical velocities higher than the sound speed at scales of $r>$1600~au. This suggests that the origin of angular momentum in the outer protostellar envelopes could be the gravitationally-driven turbulence due to large-scale collapse motions at the interface between filaments and cores, or the dissipation of the large-scale ISM turbulent cascade propagating with subsonic velocities to 1600~au envelope scales.\n\\end{enumerate}\n\n\n\n\n\\begin{acknowledgements}\nWe thank the IRAM staff for their support carrying out the CALYPSO observations. \nThis work has benefited from the support of the European Research Council under the European Union's Seventh Framework Programme (Advanced Grant ORISTARS with grant agreement no. 291294 and Starting Grant MagneticYSOs with grant agreement no. 679937). M.G. thanks the Max-Planck Institute for Radio Astronomy for its support toward the end of this work. We would like to thank Cecilia Cecarelli for comments and suggestions on the estimation of column density, and Nagayoshi Ohashi and Jaime E. Pineda for valuable discussions on the interpretation. We also thank the referee for the useful comments which helps to improve this paper.\n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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}#1)}\n\\newcommand{\\fpx}[0]{\\mathbb F_p^\\times}\n\\newcommand{\\lr}[1]{\\left(#1\\right)}\n\\newcommand{\\flr}[1]{\\left\\lfloor #1\\right\\rfloor}\n\\newcommand{\\ceil}[1]{\\left\\lceil#1\\right\\rceil}\n\\newcommand{\\supp}[0]{\\text{supp}}\n\\newcommand{\\Ker}[0]{\\text{Ker}}\n\\newcommand{\\qu}[1]{``#1''}\n\n\n\n\n\\newcommand\\ir{\\mathrm{Irr}} \n\\newcommand\\irr{\\mathrm{Irr}_1}\n\\newcommand\\lin{\\mathrm{Lin}}\n\\newcommand\\Ir{\\mathrm{I}}\n\\newcommand\\Irr{\\mathrm{I}_1} \n\\newcommand\\cd{\\mathrm{c. d.}} \n\\newcommand\\gc{\\gcd}\n\\newcommand\\be{\\begin{eqnarray*}}\n\\newcommand\\ee{\\end{eqnarray*}}\n\\newcommand\\beq{\\begin{equation}}\n\\newcommand\\eeq{\\end{equation}}\n\\newcommand{\\mathbb{E}}{\\mathbb{E}}\n\\newcommand{\\Cal}[1]{\\mathcal{#1}}\n\n\\newcommand\\ben{\\begin{eqnarray}}\n\\newcommand\\een{\\end{eqnarray}}\n\\newcommand\\ord{\\mathrm{ord}}\n\n\n\\begin{document}\n\n\n\\baselineskip=17pt\n\n\n\n\\title{On iterated product sets with shifts II}\n\n\\author[B. Hanson]{Brandon Hanson} \\address{Pennsylvania State University\\\\\nUniversity Park, PA, USA}\n\\email{bwh5339@psu.edu}\n\\author[O. Roche-Newton]{Oliver Roche-Newton} \\address{Johann Radon Institute for Computational and Applied Mathematics\\\\\nLinz, Austria}\n\\email{o.rochenewton@gmail.com}\n\\author[D. Zhelezov]{Dmitrii Zhelezov} \\address{Alfr\\'{e}d R\\'{e}nyi Institute of Mathematics \\\\ \nHungarian Academy of Sciences, Budapest, Hungary }\n\\email{dzhelezov@gmail.com}\n\\date{}\n\n\\begin{abstract} The main result of this paper is the following: for all $b \\in \\mathbb Z$ there exists $k=k(b)$ such that\n\\[ \\max \\{ |A^{(k)}|, |(A+u)^{(k)}|  \\}  \\geq |A|^b, \\]\nfor any finite $A \\subset \\mathbb Q$ and any non-zero $u \\in \\mathbb Q$. Here, $|A^{(k)}|$ denotes the $k$-fold product set $\\{a_1\\cdots a_k : a_1, \\dots, a_k \\in A \\}$.\n\n\nFurthermore, our method of proof also gives the following $l_{\\infty}$ sum-product estimate. For all $\\gamma >0$ there exists a constant $C=C(\\gamma)$ such that for any $A \\subset \\mathbb Q$ with $|AA| \\leq K|A|$ and any $c_1,c_2 \\in \\mathbb Q \\setminus \\{0\\}$, there are at most $K^C|A|^{\\gamma}$ solutions to\n\\[ c_1x + c_2y =1 ,\\,\\,\\,\\,\\,\\,\\, (x,y) \\in A \\times A.\n\\]\nIn particular, this result gives a strong bound when $K=|A|^{\\epsilon}$, provided that $\\epsilon >0$ is sufficiently small, and thus improves on previous bounds obtained via the Subspace Theorem.\n\nIn further applications we give a partial structure theorem for point sets which determine many incidences and prove that sum sets grow arbitrarily large by taking sufficiently many products.\n\n\n\n\n\\end{abstract}\n\n\n\\maketitle\n\n\\section{Introduction}\n\n\\subsection{Background and statement of main results}\n\n\nLet $A$ be a finite set of rational numbers and let $u\\in \\QQ$ be non-zero. In this article we wish to investigate the sizes of the $k$-fold product sets\n\\[A^{(k)}:=\\{a_1\\cdots a_k:a_1,\\ldots,a_k\\in A\\}\\]\nand\n\\[(A+u)^{(k)} =\\{(a_1+u)\\cdots (a_k+u):a_1,\\ldots,a_k\\in A\\}.\\]\nThis is an instance of a sum-product problem. Recall that the Erd\\H{o}s-Szemer\\'{e}di \\cite{ES} sum-product conjecture states that, for all $\\epsilon >0$ there exists a constant $c(\\epsilon)>0$ such that\n\\[ \\max \\{|A+A|, |AA| \\} \\geq c(\\eps) |A|^{2-\\eps} \\]\nholds for any $A \\subset \\mathbb Z$. Here $A+A:=\\{a+b : a,b \\in A \\}$ is the \\textit{sum set} of $A$, and $AA$ is another notation for $A^{(2)}$. Erd\\H{o}s and Szemer\\'{e}di also made the more general conjecture that for any finite $A \\subset \\mathbb Z$,\n\\[\n\\max \\{|kA|,|A^k|\\} \\geq c(\\epsilon)|A|^{k-\\epsilon},\n\\]\nwhere $kA:= \\{a_1+ \\dots +a_k : a_1,\\dots, a_k \\in A\\}$ is the \\textit{$k$-fold sum set}. Both of these conjectures are wide open, and it is natural to also consider them for the case when $A$ is a subset of $\\mathbb R$ or indeed other fields. The case when $k=2$ has attracted the most interest. See, for example, \\cite{KS}, \\cite{KS2}, \\cite{S}, \\cite{TV} and the references contained therein for more background on the original Erd\\H{o}s-Szemer\\'{e}di sum-product problem.\n\nMost relevant to our problem is the case of general (large) $k$. Little is known about the Erd\\H{o}s-Szemer\\'{e}di conjecture in this setting, with the exception of the remarkable series of work of Chang \\cite{C} and Bourgain-Chang \\cite{BC}. This culminated in the main theorem of \\cite{BC}: for all $b \\in \\mathbb R$ there exists $ k = k(b) \\in \\mathbb Z$ such that\n\\begin{equation} \\label{BCmain}\n\\max \\{|kA|,|A^k|\\} \\geq |A|^b\n\\end{equation}\nholds for any $A \\subset \\mathbb Q$. On the other hand, it appears that we are not close to proving such a strong result for $A \\subset \\mathbb R$.\n\nIn the same spirit as the Erd\\H{o}s-Szemer\\'{e}di conjecture, it is expected that an additive shift will destroy  multiplicative structure present in $A$. In particular, one expects that, for a non-zero $u$, at least one of $|A^{(k)}|$ or $|(A+u)^{(k)}|$ is large. The $k=2$ version of this problem was considered in \\cite{GS} and \\cite{JRN}. The main result of this paper is the following analogue of the Bourgain-Chang Theorem.\n\\begin{Theorem} \\label{thm:mainmain}\nFor all $b \\in \\mathbb Z$, there exists $k=k(b)$ such that for any finite set $A \\subset \\mathbb Q$ and any non-zero rational $u$,\n\\[  \\max \\{ |A^k|, |(A+u)^k| \\} \\geq |A|^b . \\]\n\\end{Theorem}\n\nThis paper is a sequel to \\cite{HRNZ}, in which the main result was the following.\n\\begin{Theorem} \\label{thm:usold} For any finite set $A \\subset \\mathbb Q$ with $|AA| \\leq K|A|$, any non-zero $u \\in \\mathbb Q$ and any positive integer $k$,\n$$| (A+u) ^{(k)}| \\geq \\frac{|A|^k}{(8k^4)^{kK}}. $$\n\\end{Theorem}\nThe proof of this result was based on an argument that Chang \\cite{C} introduced to give similar bounds for the $k$-fold sum set of a set with small product set. Theorem \\ref{thm:usold} is essentially optimal when $K$ is of the order $c\\log|A|$, for a sufficiently small constant $c=c(k)$. However, the result becomes trivial when $K$ is larger, for example if $K=|A|^{\\epsilon}$ and $\\eps>0$. The bulk of this paper is devoted to proving the following theorem, which gives a near optimal bound for the size of $(A+u)^{(k)}$ when $K=|A|^{\\eps}$,\nfor a sufficiently small but positive $\\eps$.\n\\begin{Theorem} \\label{thm:main1}\nGiven $0<\\gamma < 1\/2$, there exists a positive constant $C=C(\\gamma, k)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$ and any non-zero rational $u$,\n\\[ | (A+u) ^{(k)}|  \\geq \\frac{|A|^{k(1-\\gamma)-1}}{K^{Ck}}.\\]\n\\end{Theorem}\n\nIn fact, we prove a more general version of Theorem \\ref{thm:main1} in terms of certain weighted energies and so-called $\\Lambda$-constants (see Theorem \\ref{thm:lambda} for the general statement that implies Theorem \\ref{thm:main1} - see sections \\ref{sec:energy} and \\ref{sec:lambda} for the relevant definitions of energy and $\\Lambda$-constants). This more general result is what allows us to deduce Theorem \\ref{thm:mainmain}.\n\n\n\n\\subsection{A subspace type theorem -- an $l_\\infty$ sum-product estimate}\n\nIt appears that Theorem \\ref{thm:mainmain}, as well as the forthcoming generalised form of Theorem \\ref{thm:main1}, lead to some interesting new applications. To illustrate the strength of these sum-product results, we present three applications in this paper.\n\nOur main application concerns a variant of the celebrated Subspace Theorem by Evertse, Schmidt and Schlikewei \\cite{evertse2002linear} which, after quantitative improvements by Amoroso and Viada \\cite{amoroso2009small}, reads as follows.\n\nSuppose $a_1, \\ldots,  a_k \\in \\mathbb{C}^*$, $\\alpha_1,\\ldots,\\alpha_r \\in \\mathbb C^*$ and define\n$$\n\\Gamma = \\{\\alpha_1^{z_1} \\cdots \\alpha_r^{z_r}, z_i \\in \\mathbb{Z} \\},\n$$\nso $\\Gamma$ is a free multiplicative group\\footnote{The original theorem is formulated in a more general setting, namely for the division group of $\\Gamma$, but we will stick to the current formulation for simplicity.} of rank $r$. Consider the equation\n\\begin{equation} \\label{eq:subspace_eq}\na_1x_1 + a_2x_2 + \\cdots + a_kx_k = 1 \n\\end{equation}\nwith $a_i \\in \\mathbb{C}^*$ viewed as fixed coefficients and $x_i \\in \\Gamma$ as variables. A solution $(x_1, \\ldots, x_k)$ to (\\ref{eq:subspace_eq}) is called \\emph{nondegenerate} if\nfor any non-empty $J \\subsetneq \\{1, \\ldots, k \\}$\n$$\n\t\\sum_{i \\in J} a_ix_i \\neq 0.\n$$\n\n\\begin{Theorem}[The Subspace Theorem, \\cite{evertse2002linear} \\cite{amoroso2009small} ] \\label{thm:subspace}\nThe number $A(k, r)$ of nondegenerate  solutions to (\\ref{eq:subspace_eq}) satisfies the bound\n\\begin{equation} \\label{subspace_thm_ineq}\nA(k, r) \\leq {(8k)}^{4k^4(k + kr + 1)}.\n\\end{equation}\n\\end{Theorem} \n\n\n\n\n\n\nThe Subspace Theorem dovetails nicely to the following version of the Freiman Lemma. \n\\begin{Theorem} \\label{thm:FR-lemma}\nLet $(G, \\cdot)$ be a torsion-free abelian group and \n$A \\subset G$ with $|AA| < K|A|$. Then $A$ is contained in a subgroup $G' < G$ of rank at most $K$.\n\\end{Theorem}\n\nNow assume for simplicity that $A \\subset \\mathbb{Q}$ and $|AA| \\leq K|A|$. \n Let us call such sets (this definition generalizes of course to an arbitrary ambient group) $K$-\\emph{almost subgroups} \\footnote{One could've used a more general framework of $K$-\\emph{approximate subgroups} introduced by Tao. We decided to introduce a simpler definition in order to avoid technicalities. However, in the abelian setting the definitions are essentially equivalent.}.\n \n\nWe now show that it is natural to expect that the Subspace Theorem generalises to $K$-almost subgroups with $K$ taken as a proxy for the group rank. A straightforward corollary of Theorem~\\ref{thm:FR-lemma} and Theorem~\\ref{thm:subspace} is as follows. \n\n\\begin{Corollary}[Subspace Theorem for $K$-almost subgroups] \\label{corr:subspace_almost_subgroups}\n  Let $A$ be a $K$-almost subgroup. Then the number $A(k, K)$ of non-degenerate solutions $(x_1, x_2, \\ldots, x_k) \\in A^k$ to\n$$\nc_1x_1 + c_2x_2 + \\ldots c_kx_k = 1\n$$  \nwith fixed coefficients $c_i \\in \\mathbb{C^*}$ is bounded by \n$$ \nA(k, K) \\leq {(8k)}^{4k^4(k + kK + 1)}.\n$$\n  \n\\end{Corollary}\n\nSimilarly to Theorem~\\ref{BCmain}, the bound of Corollary~\\ref{corr:subspace_almost_subgroups} becomes trivial when $A$ is large and $K$ is larger than $c\\log |A|$ for some small $c > 0$. \n\nWe conjecture that a much stronger polynomial bound holds.\n\n\\begin{Conjecture} \\label{conj:Ksubspace}\n There is a constant $c(k)$ such that Corollary~\\ref{corr:subspace_almost_subgroups} holds with the bound\n $$\n  A(k, K) \\leq K^{c(k)}.\n $$\n\\end{Conjecture}\n\n\nWe can support Conjecture~\\ref{conj:Ksubspace} with a special case $k = 2$ and $A \\subset \\mathbb{Q}, c_i \\in \\mathbb{Q}$ and a somewhat weaker estimate, which we see as a proxy for the Beukers-Schlikewei Theorem \\cite{beukers1996equation}.\n\n\\begin{Theorem}[Weak Beukers-Schlikewei for $K$-almost subgroups] \\label{thm:BS_almost_subgroups}\n For any $\\gamma > 0$ there is $C(\\gamma) > 0$ such that for any $K$-almost subgroup $A \\subset \\mathbb{Q}$ and fixed non-zero $c_1, c_2 \\in \\mathbb{Q}$ the number $A(2, K)$ of solutions $(x_1, x_2) \\in A^2$ to \n $$\n c_1x_1 + c_2 x_2 = 1\n $$\n is bounded by \n $$\n A(2, K) \\leq |A|^\\gamma K^C.\n $$\n\n\\end{Theorem}\nOne can view Theorem \\ref{thm:BS_almost_subgroups} as an $l_{\\infty}$ version of the weak Erd\\H{o}s-Szemer\\'{e}di sum-product conjecture. The \\textit{weak Erd\\H{o}s-Szemer\\'{e}di conjecture} is the statement that, if $|AA| \\leq K|A|$ then $|A+A| \\geq K^{-C}|A|^2$ for some positive absolute constant $C$. For $A \\subset \\mathbb Z$, this result was proved in \\cite{BC}, but the conjecture remains open over the reals. \n\nA common approach to proving sum-product estimates is to attempt to show that, for a set $A$ with small product set,\nthe \\textit{additive energy} of $A$, which is defined as the quantity\n\\[E_+(A):= |\\{ (a,b,c,d) \\in A^4 : a+b=c+d \\}|, \\]\nis small. Indeed, this was the strategy implemented in \\cite{C} and \\cite{BC}, the latter of which showed\\footnote{This is something of an over-simplification, as \\cite{BC} in fact proved a much more general result which bounded the multi-fold additive energy with weights attached.} that, for all $\\gamma >0$, there is a constant $C=C(\\gamma)$ such that for any $A \\subset \\mathbb Q$ with $|AA| \\leq K|A|$,\n\\begin{equation} \nE_+(A) \\leq K^C|A|^{2+ \\gamma}. \n\\label{eq:weakES}\n\\end{equation}\nSince there are at least $|A|^2$ trivial solutions when $\\{a,b\\}=\\{c,d\\}$, this bound is close to best possible. It then follows from a standard application of the Cauchy-Schwarz inequality that\n\\[|A+A| \\geq \\frac{|A|^{2-\\gamma}}{K^C}. \\]\nDefining the representation function $r_{A+A}(c)=|\\{(a_1,a_2) \\in A \\times A : a_1+a_2=c\\}|$, it follows that\n\\[E_+(A)= \\sum_x r_{A+A}(x)^2,\\]\nand so bounds for the additive energy can be viewed as $l_2$ estimates for this representation function.\n\nTheorem \\ref{thm:BS_almost_subgroups} gives the stronger $l_{\\infty}$ estimate: it says that, if $|AA| \\leq K|A|$ then $r_{A+A}(c) \\leq K^C|A|^{\\gamma}$ for all $c \\neq 0$. This implies \\eqref{eq:weakES}, and thus in turn the weak Erd\\H{o}s-Szemer\\'{e}di sum-product conjecture. We prove Theorem~\\ref{thm:BS_almost_subgroups} in Section~\\ref{sec:conclusion}.\n\n\n\\begin{Remark}\n   It is highly probable that our method can be combined with the ideas of \\cite{bourgain2009sum} which would generalize Theorem~\\ref{thm:BS_almost_subgroups} to $K$-almost subgroups consisting of algebraic numbers of degree at most $d$ (though not necessarily contained in the same field extension). The upper power $C$ is going to depend on $d$ then, so the putative bound (using the notation of Theorem~\\ref{thm:BS_almost_subgroups}) is \n   $$\n      A(2, K) \\leq C'(d)|A|^\\gamma K^{C(\\gamma, d)}\n   $$\nwith some $C, C' > 0$.\nWe are going to consider this matter in detail elsewhere. Note, however, that proving a similar statement with no dependence on $d$ seems to be a significantly harder problem. \n\\end{Remark}\n\n\\subsection{Further applications}\n\n\\subsubsection{An inverse Szemer\\'{e}di-Trotter Theorem} Theorem \\ref{thm:BS_almost_subgroups} can be interpreted as a partial inverse to the Szemer\\'{e}di-Trotter Theorem. The Szemer\\'{e}di-Trotter Theorem states that, if $P$ is a finite set of points and $L$ is a finite set of lines in $\\mathbb R^2$, then the number of incidences $I(P,L)$ between $P$ and $L$ satisfies the bound\n\\begin{equation}\nI(P,L):= |\\{(p,l) \\in P \\times L : p \\in l \\}| =O(|P|^{2\/3}|L|^{2\/3} +|P| + |L| ).\n\\label{eq:ST}\n\\end{equation}\nThe term $|P|^{2\/3}|L|^{2\/3}$ above is dominant unless the sizes of $P$ and $L$ are rather imbalanced. The Szemer\\'{e}di-Trotter Theorem is tight, up to the multiplicative constant.  \n\nIt is natural to consider the inverse question: for what sets $P$ and $L$ is it possible that $I(P,L) = \\Omega (|P|^{2\/3}|L|^{2\/3})$? The known constructions of point sets which attain many incidences appear to all have some kind of lattice like structure. This perhaps suggests the loose conjecture that point sets attaining many incidences must always have some kind of additive structure, although such a conjecture seems to be far out of reach to the known methods.\n\nHowever, with an additional restriction that $P=A \\times A$ with $A \\subset \\mathbb Q$, Theorem \\ref{thm:mainmain} leads to the following partial inverse theorem, which states that if $A$ has small product set then $I(P,L)$ cannot be maximal.\n\n\\begin{Theorem} \\label{thm:STinverse} For all $\\gamma \\geq 0$ there exists a constant $C=C(\\gamma)$ such that the following holds. Let $A$ be a finite set of rationals such that $|AA| \\leq K|A|$ and let $P= A \\times A$. Then, for any finite set $L$ of lines in the plane, $I(P,L)\\leq 3 |P| + |A|^{\\gamma}K^C|L|$.\n\\end{Theorem}\n\nIn fact, not only does this show that $I(A \\times A,L)$ cannot be maximal when $|AA|$ is small, but better still the number of incidences is almost bounded by the trivial linear terms in \\eqref{eq:ST}. The insistence that the point set is a direct product is rather restrictive. However, since many applications of the Szemer\\'{e}di-Trotter Theorem make use of direct products, it seems likely that Theorem \\ref{thm:STinverse} could be useful. The proof is given in Section \\ref{sec:applications}.\n\n\\subsubsection{Improved bound for the size of an additive basis of a set with small product set} \n\nTheorem~\\ref{thm:BS_almost_subgroups} also yields the following application concerning the problem of bounding the size of an additive basis considered in \\cite{shkredov2016additive}. We can significantly improve the bound in the rational setting, pushing the exponent in (\\ref{eq:basis_bound}) from $1\/2 + 1\/442 - o_\\epsilon(1)$ to $2\/3 - o_\\epsilon(1)$ in the limiting case $K = |A|^{\\epsilon}$. \n\n\n\\begin{Theorem} \\label{thm:additivebasis}\nFor any $\\gamma > 0$ there exists $C(\\gamma)$ such that\nfor an arbitrary $A \\subset \\mathbb{Q}$ with $|AA| = K|A|$ and $B, B' \\subset \\mathbb{Q}$,\n$$\nS := \\left|\\{(b, b') \\in B \\times B' : b + b' \\in A\\} \\right| \\leq 2|A|^\\gamma K^C \\min \\{|B|^{1\/2}|B'| + |B|, |B'|^{1\/2}|B| + |B'| \\}. \n$$\nIn particular, for any $\\gamma > 0$ there exists $C(\\gamma)$ such that if $A \\subset B + B$ then\n\n\\begin{equation} \\label{eq:basis_bound}\n|B| \\geq |A|^{2\/3 - \\gamma}K^{-C}.\n\\end{equation}\n\n\\end{Theorem}\n\nThe proof of Theorem \\ref{thm:additivebasis} is given in Section \\ref{sec:applications}.\n\\begin{Remark}\n During the preparation of the manuscript we became aware that Cosmin Pohoata has independently proved Theorem~\\ref{thm:additivebasis} using an earlier result of Chang and by a somewhat different method.\n\\end{Remark}\n\n\\begin{comment}\n\\begin{proof}\nWe will prove that \n\\begin{equation}\nS \\leq 2|A|^\\gamma K^C(|B'|^{1\/2}|B| + |B'|).\n\\label{eq:case1}\n\\end{equation}\nSince the roles of $B$ and $B'$ are interchangeable, \\eqref{eq:case1} also implies that $S \\leq 2|A|^\\gamma K^C(|B|^{1\/2}|B'| + |B|)$, and thus completes the proof.\n\nLet $\\gamma > 0$ and $C(\\gamma)$, given by Theorem~\\ref{thm:BS_almost_subgroups}, be fixed. Without loss of generality assume that $S \\geq 2|B'|$ as otherwise the claimed bound is trivial. \n\nFor each $b \\in B$ define \n$$\nS_b := \\{ b' \\in B' : b + b' \\in A\\},\n$$\nand similarly for $b' \\in B'$\n$$\nT_{b'} := \\{b \\in B: b' + b \\in A \\}.\n$$\nIt follows from Theorem~\\ref{thm:BS_almost_subgroups} that for $b_1,b_2 \\in B$ with $b_1 \\neq b_2$ \n$$\n|S_{b_1} \\cap S_{b_2}| \\leq |A|^\\gamma K^C\n$$\nsince each $x \\in S_{b_1} \\cap S_{b_2}$ gives a solution $(a, a') := (b_1 + x, b_2 + x)$ to\n$$\na - a' = b_1 - b_2\n$$\nwith $a, a' \\in A$.\n\nOn the other hand, by double-counting and the Cauchy-Schwarz inequality,\n$$\n\\sum_{b \\in B} |S_b| + \\sum_{b_1,b_2 \\in B : b_1 \\neq b_2} |S_{b_1} \\cap S_{b_2}| =\n\\sum_{b' \\in B'} |T_{b'}|^2 \\geq |B'|^{-1}(\\sum_{b' \\in B'} |T_{b'}|)^2 = |B'|^{-1}S^2.\n$$\nTherefore,\n$$\n\\sum_{b_1,b_2 \\in B : b_1 \\neq b_2} |S_{b_1} \\cap S_{b_2}| \\geq |B'|^{-1}S^2 - \\sum_{b \\in B} |S_b| = |B'|^{-1}S^2 - S \\geq \\frac{1}{2} |B'|^{-1}S^2\n$$\nby our assumption.\n\n\nThe left-hand side is at most $ |B|^2|A|^\\gamma K^C$, and so\n$$\n\tS \\leq (2|A|^\\gamma K^C)^{1\/2} |C|^{1\/2}|B'|,\n$$\nwhich completes the proof.\n\n\n\n\\end{proof}\n\\end{comment}\n\n\n\n\n\\subsubsection{Unlimited growth for products of difference sets} It was conjectured in \\cite{BRNZ} that for any $b \\in \\mathbb R$ there exists $k=k(b) \\in \\mathbb N$ such that for all $A \\subset \\mathbb R$\n\\[|(A-A)^k| \\geq |A|^b .\\]\nIn another application of Theorem \\ref{thm:mainmain}, we give a positive answer to this question under the additional restriction that $A \\subset \\mathbb Q$. In fact, we prove the following stronger statement.\n\n\n\\begin{Theorem} \\label{thm:proddiff}\nFor any $b \\in \\mathbb R$ there exists $k=k(b) \\in \\mathbb N$ such that for all $A \\subset \\mathbb Q$ and $B \\subset \\mathbb Q$ with $|B| \\geq 2$,\n\\[|(A+B)^k| \\geq |A|^b .\\]\n\n\\end{Theorem}\nThe proof is given in Section \\ref{sec:applications}.\n\n\\subsection{The structure of the rest of this paper}\n\nIn section \\ref{sec:energy}, we introduce a new kind of mixed energy, and establish some initial bounds on this energy which are strong when the multiplicative doubling $K$ is of the order $c\\log |A|$ for a sufficiently small constant $c$. The structure of these arguments are similar to those introduced by Chang in \\cite{C}, and also used by the authors in \\cite{HRNZ}. We also introduce the notion of separating constants in section \\ref{sec:energy}, which generalises that of the aforementioned mixed energy.\n\nSection \\ref{sec:lambda} begins by stating the crucial Theorem \\ref{thm:goodsubset}, which states that is $|AA|$ is small then there is a large subset $A' \\subset A$ with a good separating constant. The rest of the section introduces the language of $\\Lambda$-constants and some of their crucial properties. These properties are then used in section \\ref{sec:conclusion} to conclude the proofs of the main results of this paper, Theorems \\ref{thm:mainmain}, \\ref{thm:main1} and \\ref{thm:BS_almost_subgroups}, using Theorem \\ref{thm:goodsubset} as a black box.\n\nIt then remains to prove Theorem \\ref{thm:goodsubset}. This is a long and technical proof, where we need to amplify the bounds obtained in section \\ref{sec:energy} in several stages. This process happens in sections \\ref{sec:fibers}, \\ref{sec:iterationscheme}, \\ref{sec:betterpair}, \\ref{sec:strongpair} and \\ref{sec:conclusion2}, and closely follows the exposition in \\cite{Z}.\\footnote{We recommend that the reader consult \\cite{Z} for more information about the proof of the Bourgain-Chang Theorem, and particularly the early parts of \\cite{Z}, where an attempt is made to outline some heuristics of the proof.} Finally, in section \\ref{sec:applications}, we give proofs of further applications of our main results.\n\n\n\n\n\n\\section{A Chang-type bound for the mixed energy} \\label{sec:energy}\n\nDifferent kinds of energies play a pivotal role in the work of Chang \\cite{C} and Bourgain-Chang \\cite{BC}, as well as \\cite{HRNZ}. In \\cite{C}, it was proved that, for any finite set of rationals $A$ with $|AA| \\leq K|A|$, the \\textit{k-fold additive energy}, which is defined as the number of solutions to\n\\begin{equation} \\label{changenergy}\na_1 + \\cdots + a_k = a_{k+1} + \\cdots a_{2k}, \\,\\,\\,\\,\\,\\,\\,\\, (a_1,\\dots,a_{2k}) \\in A^{2k}, \n\\end{equation}\nis at most $(2k^2-k)^{kK}|A|^{k}$. A simple application of the Cauchy-Schwarz inequality then implies that the \\textit{$k$-fold sum set} satisfies the bound\n\\[ |kA| \\geq \\frac{|A|^k}{(2k^2-k)^{kK}} .\\]\nBound \\eqref{changenergy} is close to optimal when $K=c \\log |A|$, but becomes trivial when $K=|A|^{\\eps}$. In \\cite{BC}, (a weighted version of) this bound was used as a foundation, and developed considerably courtesy of some intricate decoupling arguments, in order to prove a bound for the $k$-fold additive energy which remains very strong when $K$ is of the order $|A|^{\\eps}$.\n\nIn \\cite{HRNZ}, we followed a similarly strategy to that of \\cite{C}, proving that for any finite set of rationals $A$ with $|AA| \\leq K|A|$ and any non-zero rational $u$, the \\textit{k-fold multiplicative energy} of $A+u$, which is defined as the number of solutions to\n\\begin{equation}\n(a_1 + u) \\cdots  (a_k+u) = (a_{k+1} + u) \\cdots (a_{2k} +u ), \\,\\,\\,\\,\\,\\,\\,\\, (a_1,\\dots,a_{2k}) \\in A^{2k}, \n\\label{changprequel}\n\\end{equation}\nis at most $(Ck^2)^{kK}|A|^{k}$. Unfortunately, in adapting the approach of \\cite{C} in order to bound the number of solutions to \\eqref{changprequel} in \\cite{HRNZ}, we encountered some difficulties with dilation invariance which made the argument rather more complicated, and we were unable to marry our methods with those of \\cite{BC} to obtain a strong bound when $K$ is of order $|A|^{\\eps}$.\n\nIn this paper, we modify the approach of \\cite{HRNZ} by working with a different form of energy. Consider the following representation function:\n\\[r_k(x,y)=|\\{(a_1,\\ldots,a_k)\\in A^k  :a_1\\cdots a_k=x,\\ (a_1+u)\\cdots(a_k+u)=y\\}|.\\]\nThen, because $r_k$ is supported on $A^{(k)}\\times (A+u)^{(k)}$, it follows from the Cauchy-Schwarz inequality that\n\\begin{equation}\n|A|^{2k}=\\left(\\sum_{(x,y)\\in A^{(k)}\\times (A+u)^{(k)} }r_k(x,y)\\right)^2\\leq |A^{(k)}||(A+u)^{(k)}|\\sum_{(x,y)\\in A^{(k)}\\times (A+u)^{(k)}} r_k(x,y)^2.\n\\label{CSbasic}\n\\end{equation}\nThe innermost sum is the quantity\n\\[\\tilde E_k(A;u):=\\left|\\left\\{(a_1,\\ldots,a_k,b_1,\\ldots,b_k)\\in A^{2k} :\\prod_{i=1}^ka_i=\\prod_{i=1}^kb_i,\\ \\prod_{i=1}^k(a_i+u)=\\prod_{i=1}^k(b_i+u)\\right\\}\\right|.\\]\n\nWe summarise this in the following lemma.\n\\begin{Lemma} \\label{lem:CSbasic}\nFor any finite set $A\\subset \\mathbb R$, any  $u \\in \\mathbb R \\setminus \\{0\\}$ and any integer $k \\geq 2$, we have\n\\[|A|^{2k} \\leq |A^{(k)}||(A+u)^{(k)}| \\tilde E_k(A;u) .\\]\nIn particular,\n\\[\\frac{|A|^k}{\\tilde E_k(A;u)^{1\/2}}\\leq \\max\\{|A^{(k)}|,|(A+u)^{(k)}|\\}.\\]\n\\end{Lemma}\n\nOur goal is to estimate this energy and to show that, at least for sets of rationals, it cannot ever be too big. \n\n\nIn this section we seek to give an initial upper bound for $\\tilde E_k(A;u)$. The strategy is close to that of Chang \\cite{C}. There are also clear similarities with the prequel to this paper \\cite{HRNZ}.\n\nTo do this, as in \\cite{HRNZ}, we will write $\\tilde E_k(A;u)$ in terms of Dirichlet polynomials. In this case, our Dirichlet polynomials will be functions of the form \\[F(s_1,s_2)=\\sum_{(a,b)\\in\\QQ^2}\\frac{f(a,b)}{a^{s_1}b^{s_2}}\\] where $f:\\QQ^2\\to\\CC$ is some function of finite support. It will also be more convenient to count weighted energy. For $w_a$ a sequence of non-negative weights on $A$, let\n\\[\\tilde E_{k,w}(A;u)=\\sum_{\\substack{a_1\\cdots a_k=b_1\\cdots b_k\\\\ (a_1+u)\\cdots(a_k+u)=(b_1+u)\\cdots(b_k+u)}}w_{a_1}\\cdots w_{a_k}w_{b_1}\\cdots w_{b_k}\\]\n\n\\begin{Lemma} \\label{lem:direnergy}\nLet $A$ be a finite set of rational numbers and let $u$ be a non-zero rational number. Then, for any integer $k \\geq 2$, we have\n\\[\\tilde E_{k,w}(A;u)=\\lim_{T\\to \\infty}\\frac{1}{T^2}\n\\int_0^T\\int_0^T\\left|\\sum_{a\\in A} w_aa^{it_1}(a+u)^{it_2}\\right|^{2k}dt_1dt_2.\\]\n\\end{Lemma}\n\\begin{proof}\nExpanding, the double integral on the right hand side is equal to\n\\begin{multline*}\n\\sum_{a_1,\\ldots,a_k\\in A}\\sum_{b_1,\\ldots,b_k\\in A}w_{a_1}\\cdots w_{a_k}w_{b_1}\\cdots w_{b_k}\\cdot\\\\\n\\cdot\\int_0^T (a_1\\cdots a_kb_1^{-1}\\cdots b_k^{-1})^{it_1}dt_1\\int_0^T((a_1+u)\\cdots (a_k+u)(b_1+u)^{-1}\\cdots (b_k+u)^{-1})^{it_2}dt_2.\\end{multline*}\nNow\n\\[\\frac{1}{T}\\int_0^T (u\/v)^{it}dt=\\begin{cases}1&\\text{ if }u=v,\n\\\\ O_{u,v}(T^{-1})&\\text{ if }u\\neq v.\\end{cases}\\]\nFrom this, the lemma follows.\n\\end{proof}\n\nLet $\\|\\cdot \\|_{2k}$ be the standard norm in $L^{2k}[0, T]^2$, normalised such that $\\| 1\\|_{2k} = 1$. So,\n\\[\n\\| f \\|_{2k} := \\left( \\frac{1}{T^2} \\int_0^T \\int_0^T|f(t)|^{2k} dt \\right)^{1\/2k}. \n\\]\n\n\\begin{Lemma} \\label{lem:split}\nLet $\\cJ$ be a set of integers and decompose it as $\\cJ=\\cJ_1 \\cup \\cdots \\cup\\cJ_N$. For each $j \\in \\cJ$ let $f_j:\\RR\\times \\RR\\to \\CC$ be a function belonging to $L^{2k}\\lr{\\RR^2}$ for every integer $k\\geq 2$. Then, for every integer $k \\geq 2$,\n\\begin{multline}\\label{split}\\lim_{T\\to \\infty}\\lr{\\frac{1}{T^2}\\int_0^T\\int_0^T\\left|\\sum_{j\\in\\cJ} f_j(t_1,t_2)\\right|^{2k}dt_1dt_2}^{1\/k} \\\\\\leq N\\sum_{n=1}^N \\lim_{T\\to \\infty}\\lr{\\frac{1}{T^2}\\int_0^T\\int_0^T\\left| \\sum_{j\\in \\cJ_n}f_j(t_1,t_2)\\right|^{2k}dt_1dt_2}^{1\/k}.\n\\end{multline}\n\\end{Lemma}\n\n\\begin{proof} \nIt suffices to prove the inequality for all sufficiently large $T$, which we assume fixed for now.\nThen\n\\begin{equation} \\label{eq:Lknormsum}\n\\lr{\\frac{1}{T^2}\\int_0^T\\int_0^T\\left|\\sum_{j\\in\\cJ} f_j(t_1,t_2)\\right|^{2k}dt_1dt_2}^{1\/k} =\n \\left(\\left\\| \\sum_{n=1}^N\\sum_{j \\in \\cJ_n}  f_j \\right\\|_{2k} \\right)^2\\leq\\left(\\sum_{n=1}^N\\left\\| \\sum_{j \\in \\cJ_n}  f_j \\right\\|_{2k} \\right)^2,\n\\end{equation}\nby the triangle inequality. By the Cauchy-Schwarz inequality, (\\ref{eq:Lknormsum}) is bounded by \n\\begin{equation}\nN\\sum_{n=1}^N\\left\\| \\sum_{j \\in \\cJ_n}  f_j \\right\\|_{2k}^2.\n\\end{equation}\nLetting $T \\to \\infty$ we get the claim of the lemma.\n\\end{proof}\n\\begin{Corollary}\\label{Split}\nLet $A$ be a finite set of rational numbers, partitioned as $A=A_1\\cup\\cdots\\cup A_N$, let $w$ be a set of non-negative weights, and let $u$ be a non-zero rational number. Then for any integer $k \\geq 2$\n\\[\\tilde E_{k,w}(A;u)^{1\/k}\\leq N\\sum_{j=1}^N\\tilde E_{k,w}(A_j;u)^{1\/k}.\\]\n\\end{Corollary}\nNow let $p$ be a fixed prime. For $a \\in \\mathbb Q$, let $v_p(a)$ denote the $p$-adic valuation of $a$. For a set $A$ of rational numbers and an integer $t$, we let \n$A_t=\\{a\\in A:v_p(a)=t\\}$.\n\n\\begin{Lemma} \\label{thm:basecase}\nLet $p$ be a prime number. Suppose $A$ is a finite set of rational numbers and let $u$ be a non-zero rational number.\nThen for any $w$, a set of non-negative weights on $A$, and any integer $k \\geq 2$,\n\\[\\tilde E_{k,w}(A;u)^{1\/k}\\leq 2\\binom{2k}{2}\\sum_{d\\in \\ZZ}\\tilde E_{k,w}(A_d;u)^{1\/k}.\\]\n\\end{Lemma}\n\n\n\\begin{proof}\nFirst, let $A=A_+\\cup A_-$ where $A_+=\\{a\\in A:v_p(a)\\geq v_p(u)\\}$ and $A_-=\\{a\\in A:v_p(a)<v_p(u)\\}$. By Corollary \\ref{Split}, we have\n\\begin{equation}\n\\tilde E_{k,w}(A;u)^{1\/k}\\leq 2\\tilde E_{k,w}(A_+;u)^{1\/k}+2\\tilde E_{k,w}(A_-;u)^{1\/k}.\n\\label{+-split}\n\\end{equation}\nThese two terms will be dealt with in turn, starting with $E_{k,w}(A_+;u)^{1\/k}$. \n\\begin{comment}Suppose we have a solution\n\\[(a_1+u)\\cdots(a_k+u)=(b_1+u)\\cdots(b_k+u)\\]\nand so\n\\begin{equation}\n(a_1u^{-1}+1)\\cdots(a_ku^{-1}+1)=(b_1u^{-1}+1)\\cdots(b_ku^{-1}+1).\n\\label{eq:normal}\n\\end{equation}\nWe claim that $v_p$ cannot have a unique minimal value among the $a_i$ and $b_i$. Indeed, since $v_p(a_iu^{-1})\\geq 0$ and $v_p(b_iu^{-1})\\geq 0$, expanding out both sides of \\eqref{eq:normal} and simplifying\n\\[u^{-1}(a_1+\\cdots+a_k)+\\text{higher terms}=u^{-1}(b_1+\\cdots+b_k)+\\text{higher terms},\\] \nand if $v_p$ achieves a unique minimum, then the left hand side and the right hand side are divisible by distinct powers of $p$, a contradiction.\n\\end{comment}\nTo do this, we first set up some more notation. For an integer $d$, define the function\n\\[f_d(t_1,t_2):= \\sum_{a \\in A_d} w_a a^{it_1} (a+u)^{it_2} .\\]\nThen, by Lemma \\ref{lem:direnergy}\n\\[\\tilde E_{k,w}(A_+;u) = \\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T \\left|\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right|^{2k} dt_1 dt_2 .\\]\nExpanding this expression, as in the proof of Lemma \\ref{lem:direnergy}, we obtain that $\\tilde E_{k,w}(A_+;u)$ is equal to\n\\begin{equation}\n\\sum_{d_1,\\dots,d_{2k} \\geq v_p(u)}\\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  f_{d_1}(t_1,t_2)\\cdots f_{d_k}(t_1,t_2) \\bar{f_{d_{k+1}}(t_1,t_2)}\\cdots \\bar{f_{d_{2k}}(t_1,t_2)} dt_1 dt_2 .\n\\label{eq:long}\n\\end{equation}\n\nFor fixed $d_1, \\dots, d_{2k}$, the quantity\n\\[\\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  f_{d_1}(t_1,t_2)\\cdots f_{d_k}(t_1,t_2) \\bar{f_{d_{k+1}}(t_1,t_2)}\\cdots \\bar{f_{d_{2k}}(t_1,t_2)} dt_1 dt_2 .\\]\ngives a weighted count of the number of solutions to the system of simultaneous equations\n\\begin{align}\na_1 \\cdots a_k &= a_{k+1} \\cdots a_{2k} \\label{basiceq1}\n\\\\ (a_1+u) \\cdots (a_k+u) &= (a_{k+1}+u) \\cdots (a_{2k}+u),\n\\label{basiceq2}\n\\end{align}\nsuch that $a_i \\in A_{d_i}$.\n\nWe claim that there are no solutions to \\eqref{basiceq2}, and thus also no solutions to the above system, if all of the $d_i$ are distinct. Indeed, suppose we have a solution\n\\[(a_1+u)\\cdots(a_k+u)=(a_{k+1}+u)\\cdots(a_{2k}+u)\\]\nand so\n\\begin{equation}\n(a_1u^{-1}+1)\\cdots(a_ku^{-1}+1)=(b_{k+1}u^{-1}+1)\\cdots(b_{2k}u^{-1}+1).\n\\label{eq:normal}\n\\end{equation}\n\nSince $v_p(a_iu^{-1})\\geq 0$, expanding out both sides of \\eqref{eq:normal} and simplifying gives\n\\begin{equation}\nu^{-1}(a_1+\\cdots+a_k)+\\text{higher terms}=u^{-1}(b_{k+1}+\\cdots+b_{2k})+\\text{higher terms}.\n\\label{eq:contradiction}\n\\end{equation} \nIf all of the $d_i$ are distinct, then there is some unique smallest $d_i$, and thus a unique smallest value of $v_p(a_i)$. But then the left hand side and the right hand side are divisible by distinct powers of $p$, a contradiction.\n\nSo returning to \\eqref{eq:long}, we need only consider the cases in which one or more of the $d_i$ are repeated. There are three kinds of ways in which this can happen.\n\n\\begin{enumerate}\n\\item $d_i=d_i'$ with $1 \\leq i \\leq k$ and $k+1 \\leq i' \\leq 2k$. There are $k^2$ possible positions for such a pair $(i,i')$,\n\\item $d_i=d_i'$ with $1 \\leq i, i' \\leq k$. There are $\\binom{k}{2}$ possible positions for such a pair $(i,i')$,\n\\item $d_i=d_i'$ with $k+1 \\leq i, i' \\leq 2k$. There are $\\binom{k}{2}$ possible positions for such a pair $(i,i')$.\n\\end{enumerate}\n\nSuppose we are in situation (1) above. Specifically, suppose that $d_1=d_{2k}$. The other $k^2-1$ cases can be dealt with by the same argument. Then these terms in \\eqref{eq:long} can be rewritten as\n\n\\begin{multline}\n\\sum_{d_1 \\geq v_p(u)}\\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  f_{d_1}(t_1,t_2)  \\bar{f_{d_1}(t_1,t_2)} \n\\\\ \\sum_{d_2,\\dots,d_{2k-1} \\geq v_p(u)} f_{d_2}(t_1,t_2) \\cdots f_{d_k}(t_1,t_2) \\bar{f_{d_{k+1}}(t_1,t_2)}\\cdots \\bar{f_{d_{2k-1}}(t_1,t_2)} dt_1 dt_2 \n\\\\= \\sum_{d \\geq v_p(u)}\\lim_{T \\rightarrow \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  |f_{d}(t_1,t_2)   |^2 \\left |\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right |^{2(k-1)} dt_1 dt_2.\n\\end{multline}\n\nSuppose we are in situation (2). Specifically, suppose that $d_1=d_{2}$. The other $ \\binom{k}{2}-1$ cases can be dealt with by the same argument. Then these terms in \\eqref{eq:long} can be rewritten as\n\n\\begin{align*} & \\sum_{d_1 \\geq v_p(u)}  \\lim_{T\\to \\infty} \\frac{1}{T^2} \\int_0^T \\int_0^T  f_{d_1}^2(t_1,t_2)  \\sum_{d_3,\\dots, d_{2k} \\geq v_p(u)} f_{d_3}(t_1,t_2) \\cdots f_{d_k}(t_1,t_2)  \\bar{f_{d_{k+1}}(t_1,t_2)} \\cdots \\bar{f_{d_{2k}}(t_1,t_2)} dt_1 dt_2 \n\\\\& \\leq  \\sum_{d \\geq v_p(u)} \\lim_{T\\to \\infty}\\frac{1}{T^2}\\int_0^T\\int_0^T   |f_d(t_1,t_2)|^2\\left|\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right|^{k-2} \\left|\\sum_{d} \\bar{f_d(t_1,t_2)}\\right|^{k}  dt_1dt_2\n\\\\&= \\sum_{d \\geq v_p(u)} \\lim_{T\\to \\infty}\\frac{1}{T^2}\\int_0^T \\int_0^T   |f_d(t_1,t_2)|^2\\left|\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right|^{2(k-1)}   dt_1dt_2.\n\\end{align*}\n\nThe same argument also works in case (3). Returning to \\eqref{eq:long}, we then have\n\\begin{align*}\n\\tilde E_{k,w}(A_+;u)& \\leq \n \\binom{2k}{2}\\sum_{d \\geq v_p(u)} \\lim_{T\\to \\infty}\\frac{1}{T^2}\\int_0^T \\int_0^T   |f_d(t_1,t_2)|^2\\left|\\sum_{d \\geq v_p(u)} f_d(t_1,t_2)\\right|^{2(k-1)}   dt_1dt_2\\\\\n&\\leq \\binom{2k}{2}\\sum_{d\\geq v_p(u)}\\tilde E_{k,w}(A_d;u)^{1\/k}E_{k,w}(A_+;u)^{1-1\/k},\n\\end{align*}\nthe last inequality being H\\\"older's. It therefore follows that\n\\begin{equation}\n\\tilde E_{k,w}(A_+;u)^{1\/k} \\leq \\binom{2k}{2} \\sum_{d\\geq v_p(u)}\\tilde E_{k,w}(A_d;u)^{1\/k}  .\n\\label{plusbound}\n\\end{equation}\nNow we proceed to $E_{k,w}(A_-;u)^{1\/k}$. For any solution to the pair of equations\n\\begin{align*}\na_1\\cdots a_k&=a_{k+1}\\cdots a_{2k}\\\\ (a_1+u)\\cdots (a_k+u)&=(a_{k+1}+u)\\cdots (a_{2k}+u)\n\\end{align*}\nwe have a solution to the equation\n\\[(1+ua_1^{-1})\\cdots (1+ua_k^{-1})=(1+ua_{k+1}^{-1})\\cdots (1+ua_{2k}^{-1}).\\]\nAgain, we expand and simplify, using this time that $v_p(ua_i^{-1})$ is positive, and get\n\\[u(a_1^{-1}+\\cdots a_k^{-1})+\\text{higher terms}=u(a_{k+1}^{-1}+\\cdots a_{2k}^{-1})+\\text{higher terms}.\\]\nAs in the previous case\n\\footnote{Note that here we have used the information that $a_1 \\cdots a_k=a_{k+1} \\cdots a_{2k}$, whereas we did not use this when bounding $\\tilde E_{k,w}(A_+;u)$.}\n, we cannot have a unique smallest $v_p(ua_i^{-1})$. We can therefore repeat the arguments that gave us \\eqref{plusbound} in order to deduce that\n\\begin{equation}\n\\tilde E_{k,w}(A_-;u)^{1\/k} \\leq \\binom{2k}{2} \\sum_{d < v_p(u)}\\tilde E_{k,w}(A_d;u)^{1\/k}  .\n\\label{minusbound}\n\\end{equation}\nInserting \\eqref{plusbound} and \\eqref{minusbound} into \\eqref{+-split} completes the proof.\n\\end{proof}\n\n\n\nNext, this is used as a base case to give an analogous result with more primes.\n\n\\begin{Lemma} \\label{thm:chang1}\nLet $p_1,\\dots,p_K$ be a prime numbers. Suppose $A$ is a finite set of rational numbers and let $u$ be a non-zero rational number. For a vector $\\dd=(d_1,\\dots,d_K)$, define\n\\[A_{\\dd}=\\{a\\in A:v_{p_1}(a)=d_1, \\dots, v_{p_k}(a)=d_k\\}.\\]\nThen for any $w$, a set of non-negative weights on $A$, and for any integer $k \\geq 2$,\n\\[\\tilde E_{k,w}(A;u)^{1\/k}\\leq \\left(2\\binom{2k}{2} \\right)^K\\sum_{\\dd\\in \\ZZ^K}\\tilde E_{k,w}(A_{\\dd};u)^{1\/k}.\\]\n\\end{Lemma}\n\n\\begin{proof}\n\n\nThe aim is to prove that\n\\begin{multline}\n\\lim_{T\\to \\infty}\\left (\\frac{1}{T^2}\\int_0^T \\int_0^T\\left|\\sum_{\\dd\\in \\mathbb Z^K} \\sum_{a \\in A_{\\dd}}w_a a^{it_1}(a+u)^{it_2} \\right|^{2k}dt_1 dt_2 \\right)^{1\/k}\n\\\\ \\leq \\left(2\\binom{2k}{2} \\right)^K\\sum_{\\dd \\in \\mathbb Z^K}\\lim_{T\\to \\infty}\\lr{\\frac{1}{T^2}\\int_0^T \\int_0^T\\left| \\sum_{a \\in A_{\\dd}} w_a a^{it_1}(a+u)^{it_2}    \\right|^{2k}dt_1 dt_2}^{1\/k}.\n\\end{multline}\n\nWe proceed by induction on $K$, the base case $K=1$ being given by Lemma \\ref{thm:basecase}. Then\n\\begin{align*}\n&\\lim_{T\\to \\infty} \\left(\\frac{1}{T^2} \\int_0^T \\int_0^T \\left|\\sum_{\\dd \\in \\mathbb Z^K }\\sum_{a \\in A_{\\dd}}w_a a^{it_1}(a+u)^{it_2} \\right|^{2k} dt_1dt_2\\right)^{1\/k}\n\\\\&=\\lim_{T\\to \\infty} \\left(\\frac{1}{T^2} \\int_0^T \\int_0^T  \\left|\\sum_{d_K\\in \\mathbb Z} \\left(\\sum_{\\dd'\\in \\mathbb Z^{K-1} }\\sum_{a \\in A_{(\\dd',d)}}w_a a^{it_1}(a+u)^{it_2} \\right) \\right|^{2k} dt_1 dt_2\\right)^{1\/k}\n\\\\&\\leq 2\\binom{2k}{2}  \\sum_{d_K\\in \\mathbb Z} \\lim_{T\\to \\infty} \\left(\\frac{1}{T^2} \\int_0^T \\int_0^T \\left|\\sum_{\\dd'\\in \\mathbb Z^{K-1} }\\sum_{a \\in A_{(\\dd',d)}}w_a a^{it_1}(a+u)^{it_2}  \\right|^{2k} dt_1 dt_2\\right)^{1\/k}\n\\\\& \\leq 2\\binom{2k}{2}  \\sum_{d_K\\in \\mathbb Z} \\left( 2\\binom{2k}{2}  \\right)^{K-1} \\sum_{\\dd'\\in \\mathbb Z^{K-1} } \\lim_{T\\to \\infty} \\left(\\frac{1}{T^2} \\int_0^T \\int_0^T \\left| \\sum_{a \\in A_{(\\dd',d)}}w_a a^{it_1}(a+u)^{it_2}  \\right|^{2k} dt_1 dt_2\\right)^{1\/k}\n\\\\&=\\left( 2\\binom{2k}{2}  \\right)^{K} \\sum_{\\dd \\in \\mathbb Z^K}\\lim_{T\\to \\infty}\\lr{\\frac{1}{T^2}\\int_0^T \\int_0^T\\left| \\sum_{a \\in A_{\\dd}}w_a a^{it_1}(a+u)^{it_2}  \\right|^{2k}dt_1 dt_2}^{1\/k}.\n\\end{align*}\nThe first inequality above follows from an application of Lemma \\ref{thm:basecase}. The second inequality follows from the induction hypothesis.\n\\end{proof}\n\n\n\n\n\n\n\\begin{comment}\nIn particular, this implies the following version of Chang's Theorem, giving a strong estimate for the energy $\\tilde E_{k,w}(A)$ in the case when the multiplicative doubling constant $K$ is of size $c \\log |A|$.\n\n\\begin{Theorem} \\label{thm:chang} Let $A \\subset \\mathbb Q$ be finite with $|AA|=K|A|$. Then, for any non-zero rational\n\n\\end{Theorem}\n\\end{comment}\n\n\\subsection{Separating constants} \\label{sec:separatingconstants}\n\nThe following definition, which follows the terminology used in \\cite{Z}, is central to this paper. Let $\\psi$ be an arbitrary real number. A set $X \\subset \\mathbb Q$ is said\\footnote{Strictly speaking, we should perhaps include $k$ in this definition and say that a set is $(\\psi,k)$-separating if it satisfied the stated conditions. In order to simplify the notation we do not do this. Instead, we can think of $k \\geq 2$ as a fixed integer throughout the remainder of the paper, unless stated otherwise. Henceforth, this condition on $k$ will be omitted from statements of results.} to be \\textit{$\\psi$-separating} if for any non-zero $u \\in \\mathbb Q$, any set finite $Z \\subset \\mathbb Q$ of the form\n\\[Z=\\bigcup_{x \\in X} xY_x \\]\nsuch that $(x,Y_{x'}) =1$ for all $x,x' \\in X$, and any set of weights $w$ on $Z$\n\\[\\tilde E_{k,w}(Z;u)^{1\/k}\\leq \\psi \\sum_{x \\in X}\\tilde E_{k,w}(xY_x;u)^{1\/k}.\\]\n\nA first observation about separating constants comes in the form of the following claim.\n\\begin{Claim} \\label{cor:trivialseparation}\nAny finite $A \\subset \\mathbb Q$ is $|A|$-separating.\n\\end{Claim}\nThis claim follows immediately from Corollary \\ref{Split}. Combining this new definition with Lemma \\ref{thm:basecase}, we can also record the following corollary.\n\n\\begin{Corollary} \\label{cor:oneprimeseparation}\nLet $p$ be a prime number. Suppose $A$ is of the form $A=\\{p^h : h \\in H \\}$ for some finite set $H \\subset \\mathbb Z$. Then $A$ is $2 \\binom{2k}{2}$-separating.\n\\end{Corollary}\n\nWith this definition of the separating constant, we can use Lemma \\ref{thm:chang1} to get a first bound for the separating constant of a set with small product set. Once again, this bound is good when $K \\leq c \\log |A|$ for a sufficiently small constant $c$. \n\nTo do this, we recall an argument of Chang \\cite{C} which uses Freiman's Lemma to show that a set of rationals with small product set is determined by a small number of prime factors. Let $A$ be a set of rationals and let \n\\[\\cP:= \\{p: p\\text{ is prime and there exists }  a \\in A, v_p(a) \\neq 0\\} = \\{p_1,\\ldots,p_t\\}\\] \nbe the set of primes dividing some element of $A$. Abusing notation slightly, we define a map $\\cP:A\\to\\ZZ^t$ where $\\cP(a)=(v_{p_1}(a),\\ldots,v_{p_t}(a))$. Denoting by $\\cP(X)$ the image of a set $X$ under $\\cP$, observe that $\\cP(AA)=\\cP(A)+\\cP(A)$. We define the \\textit{multiplicative dimension} of $A \\subset Q$ to be the least dimension of an affine space $L$ containing $\\cP(A)$.\n\n\\begin{Theorem}[Freiman's Lemma] \\label{thm:FreimanLemma}\nLet $A\\subset \\RR^m$ be a finite set not contained in a proper affine subspace. Then \\[|A+A|\\geq (m+1)|A|-O_m(1).\\]\n\\end{Theorem}\n\n\\begin{Theorem} \\label{thm:chang2} Let $A \\subset \\mathbb Q$ be finite with $|AA|=K|A|$. Then, $A$ is $\\left(2\\binom{2k}{2} \\right)^K$-separating. \n\\end{Theorem}\n\n\\begin{proof} It follows from Freiman's Lemma that if $|AA|\\leq K|A|$ with $|A|$ sufficiently large, then $A$ has multiplicative dimension at most $K$.\n\nThis means that there is a set of $\\{p_1,\\dots,p_K\\}$ of primes and a set of vectors $\\cJ \\subset \\mathbb Z^K$ such that\n\\[A= \\bigcup_{\\jj=(j_1,\\dots,j_K) \\in \\cJ} p_1^{j_1}\\cdots p_K^{j_K} x_{\\jj}, \\]\nwhere each $x_{\\jj}$ is a rational number coprime\\footnote{We say that two rational numbers $a$ and $b$ are comprime if at least one of $v_p(a)$ and $v_p(b)$ is zero for all prime $p$. As with the case of integers, we write $(a,b)=1$.} to $p_1\\cdots p_K$. For $\\jj=(j_1,\\dots,j_K) \\in \\cJ$, write $a_{\\jj}=p_1^{j_1}\\cdots p_K^{j_K} x_{\\jj}$.\n\nNow, let\n\\[Z= \\bigcup_{\\jj \\in \\cJ} a_{\\jj}Y_{\\jj}  \\subset \\mathbb Q \\]\nwith the $(Y_{\\jj}, a_{\\jj'})=1$ for all $\\jj,\\jj' \\in \\cJ$. In particular, $Y_{\\jj}$ is coprime to $p_1\\cdots p_K$. Therefore, in the notation of Lemma \\ref{thm:chang1}\n\\[Z_{\\jj}= a_{\\jj} Y_{\\jj}. \\]\nThen, by Lemma \\ref{thm:chang1},\n\\[\\tilde E_{k,w}(Z;u)^{1\/k}\\leq \\left(2\\binom{2k}{2} \\right)^K\\sum_{\\jj\\in \\ZZ^K}\\tilde E_{k,w}(a_{\\jj}Y_{\\jj};u)^{1\/k}.\\]\n\n\n\\end{proof}\n\nWe recall now the Pl\\\"{u}nnecke-Ruzsa Theorem. See \\cite{P} for a simple inductive proof. Following convention, we state it using additive notation, although it will be used in the multiplicative setting.\n\n\\begin{Theorem} \\label{thm:Plun} Let $A$ be a subset of a commutative additive group $G$ with $|A+A| \\leq K|A|$. Then for any $h \\in \\mathbb N$,\n\\[ |hA| \\leq K^h|A|. \\]\n\\end{Theorem}\n\nOne may think of the separating constant of $X$ as a generalisation of the notion of the mixed energy $\\tilde E_k(X;u)$. Indeed, if $X$ is $\\psi$-separating then take $Y_x=\\{1\\}$ for all $x \\in X$ and $w(x)=1$ for all $x \\in X$. Then it follows that $\\tilde E_k(X;u) \\leq \\psi^k|X|^k$. In particular, Theorem \\ref{thm:chang2} implies the following result.\n\n\\begin{Theorem} \\label{thm:chang3} Let $A$ be a finite set of rational numbers and let $u \\in \\mathbb Q$ be non-zero. Suppose that $|AA| \\leq K|A|$. Then, for any integer $k \\geq 2$,\n\\[|(A+u)^k| \\geq \\frac{|A|^{k-1}}{K^k\\left( 2 \\binom{2k}{2} \\right)^{Kk}}\n\\]\n\\end{Theorem}\n\n\\begin{proof}\nBy Theorem \\ref{thm:chang2}, \n\\[ \\tilde E_k(A;u) \\leq \\left(2\\binom{2k}{2} \\right)^{Kk}|A|^k.\\] \nAlso, by Theorem \\ref{thm:Plun}, $|A^{(k)}| \\leq K^k|A|$. Inserting these two bounds into Lemma \\ref{lem:CSbasic} completes the proof.\n\\end{proof}\n\nA stronger version of Theorem \\ref{thm:chang3} was the main result of \\cite{HRNZ}, which used the standard $k$-fold multiplicative energy of the set $A+u$. \n\nA key goal of this paper is to amplify this approach in order to give a good bound for the case when $K=|A|^{\\eps}$. The advantage of working with this generalised notion of energy is that it has a crucial ``chaining property\" which will be important in the forthcoming analysis for pushing to get results for larger $K$.\n\n\\begin{Lemma} \\label{lem:chain} Let $A$ be a finite set of rationals which can be decomposed as a disjoint union\n\\[A= \\bigcup_{b \\in B} bC_b\\]\nand with $(b,C_{b'})=1$ for all $b,b' \\in B$. Assume also that $B$ is $\\psi_1$-separating and that each $C_b$ is $\\psi_2$-separating. Then $A$ is $\\psi_1\\psi_2$-separating.\n\\end{Lemma}\n\n\\begin{proof}\nLet $Z$ be a set of rationals which decomposes as\n\\[Z = \\bigcup_{a \\in A} aY_a \\]\nwith $(a,Y_{a'})=1$ for all $a,a' \\in A$. Then for any $u \\in \\mathbb Q$ and weights $w$ on $Z$,\n\\[ \\tilde E_{w,u}(Z;u)^{1\/k} = \\tilde E_{w,u} \\left (\\bigcup_{b \\in B} b \\left ( \\bigcup_{c \\in C_b} cY_{bc} \\right ) ;u \\right )^{1\/k} \\leq  \\psi_1 \\sum_{b \\in B} \\tilde E_{w,u} \\left ( b \\left ( \\bigcup_{c \\in C_b} cY_{bc} \\right );u \\right )^{1\/k}. \\]\nIn the inequality above we have used the fact that $B$ is $\\psi_1$ separating and that \n\\[ \\left (b,\\bigcup_{c \\in C_{b'}} cY_{b'c}\\right )=1\\]\nfor any $b,b' \\in B$. Indeed, take an arbitrary product $cy$ with $c \\in C_{b'}$ and $y \\in Y_{b'c}$. Then $b$ is coprime to $c$ by the hypothesis of the lemma. Also, $y$ is coprime to each element of $A$ by the definition of $Z$, which implies that $y$ is coprime to $b$ by the hypothesis of the lemma.\n\nWe therefore have\n\\begin{align*} \\tilde E_{w,u}(Z;u)^{1\/k} \\leq \\psi_1 \\sum_{b \\in B} \\tilde E_{w,u}  \\left ( \\bigcup_{c \\in C_b} c(bY_{bc}) ;u \\right ) ^{1\/k} & \\leq \\psi_1 \\sum_{b \\in B} \\psi_2 \\sum_{c \\in C_b} \\tilde E_{w,u}  \\left (  c(bY_{bc}) ;u \\right ) ^{1\/k}\n\\\\&= \\psi_1 \\psi_2 \\sum_{a \\in A} \\tilde E_{w,u}  \\left (  aY_a ;u \\right ) ^{1\/k}.\n\\end{align*}\nThe inequality above uses fact that $C_b$ is $\\psi_2$-separating and that $(c,bY_{bc'})=1$ for any $c,c' \\in C_b$. This can be verified in the same way as the previous inequality.\n\\end{proof}\n\n\\section{Lambda-constants} \\label{sec:lambda}\n\nWe will soon begin the process of amplifying Theorem \\ref{thm:chang2} from the previous section in order to get a better separating factor which leads to strong bound when $K=|A|^{\\eps}$. At the conclusion of this process we will prove the following result.\n\n\\begin{Theorem} \\label{thm:goodsubset} Given $0<\\tau, \\gamma < 1\/2$, there exist positive constants $C_1=C_1(\\tau,\\gamma,k)$ and $C_2=C_2(\\tau,\\gamma,k)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$, there exists $A' \\subset A$ with $|A'| \\geq K^{-C_1}|A|^{1-\\tau}$ such that $A'$ is $K^{C_2}|A|^{\\gamma}$-separating.\n\\end{Theorem}\n\nIn fact, one can check that the proof of Theorem \\ref{thm:goodsubset} goes through in a more general setting. Let $S: 2^{\\mathbb{Q}} \\to \\mathbb{R}$ be a function defined on rational sets with the following properties:\n\n\\begin{enumerate}\n   \\item (Trivial bound) For an arbitary set $A \\subset \\mathbb{Q}$ \n   $$\n   S(A) \\leq |A|.\n   $$\n   \n   \\item (Stability) If $A' \\subset A$ then\n   $$\n   S(A') \\leq S(A).\n   $$\n   \n   \\item ($p$-adic separation) There is an absolute constant $s \\geq 0 $ such that for any prime $p$ and $I \\subset \\mathbb{Z}$,\n   \n   $$\n    S(\\bigcup_{i \\in I} p^i) \\leq s.\n   $$\n   \n   \\item (Nesting) Let $A \\subset \\mathbb{Q}$ and $\\{ B_a \\}_{a \\in A}$ is a collection of sets such that $(a, B_{a'}) = 1$ for any $a, a' \\in A$. Further assume that $aB_a, a \\in A$ are pairwise disjoint. \n   Then\n   $$\n   S(\\bigcup_{a \\in A} aB_a) \\leq S(A) \\max_{a \\in A} S(aB_a).\n   $$\n\\end{enumerate}\n\nNote that our definition of the separating constant satisfies (1)-(4). \n\n\\begin{Theorem} \\label{thm:goodsubset_general} Let $S$ be a function with the properties above. Given $0<\\tau, \\gamma < 1\/2$, there exist positive constants $C_1=C_1(\\tau,\\gamma, s)$ and $C_2=C_2(\\tau,\\gamma, s)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$, there exists $A' \\subset A$ with $|A'| \\geq K^{-C_1}|A|^{1-\\tau}$ such that \n$$\nS(A') \\leq K^{C_2}|A|^{\\gamma}.\n$$\n\\end{Theorem}\n\nThe proof of Theorem~\\ref{thm:goodsubset}  is essentially borrowed from \\cite{BC}. We present here a  proof adapted to our setting to make the paper self-contained. The same proof applies to Theorem~\\ref{thm:goodsubset_general} with cosmetic modifications, but we expect that it might be useful to have such a general `black-box' version for future use.\n\nBefore we begin the lengthy proof of Theorem \\ref{thm:goodsubset}, we will take some time to see how it implies the two main theorems of this paper. To do this, it will be convenient to use the language of \\textit{$\\Lambda$-constants}, and to introduce some of their key properties. The main motivation behind  $\\Lambda$-constants is the stability property given by the forthcoming Corollary~\\ref{corr:stability}, which is absent in the non-weighted version of the energy. \n\nWe also encourage the interested reader to consult our preceding paper \\cite{HRNZ} for a slightly more gentle introduction to $\\Lambda$-constants in the setting of Dirichlet polynomials and more in-depth motivation behind this concept. \n\n\nLet $A \\subset \\mathbb Q$ be a finite set and let $u$ be a non-zero rational. Define\n\\[\\Lambda_k(A;u):= \\max \\tilde E_{k,w}(A;u)^{1\/k},\\]\nwhere the maximum is taken over all weights $w$ on $A$ such that\n\\begin{equation}\n\\sum_{a \\in A} w(a)^2 =1. \n\\label{eq:weightsnorm}\n\\end{equation}\nAn equivalent definition is\n\\[\\Lambda_k(A;u):= \\max \\lim_{T\\to \\infty}\n\\left \\|\\sum_{a\\in A}w_aa^{it_1}(a+u)^{it_2} \\right \\|_{2k}^2.\\]\nwhere the maximum is taken over the same range of weights.\n\n\n\n\\begin{Lemma} \\label{lm:anyweights}\n Let $A \\subset \\mathbb Q$ be a finite set with some non-negative real weights $w_a$ assigned to each element $a \\in A$ and let $u$ be a non-zero rational. Then\n \\begin{equation} \\label{eq:lambdaanyweights}\n \\left \\| \\sum_{a \\in A} w_a a^{it_1}(a+u)^{it_2} \\right\\|^2_{2k} \\leq \\Lambda_k(A;u) \\left(\\sum_{a \\in A} w_a^2 \\right) + o_{T\\to\\infty}(1).\n \\end{equation}\n\\end{Lemma}\n\\begin{proof}\nIf $\\sum_{a \\in A} w^2_a = 0$ the claim of the lemma is trivial. Otherwise, define new weights\n\\[\nw'_a := \\frac{w_a}{(\\sum_{a \\in A} w_a^2)^{1\/2}}\n\\]\nwhich satisfy (\\ref{eq:weightsnorm}). It thus suffices to show that\n\\[\n\\left \\| \\sum_{a \\in A} w'_a a^{it_1}(a+u)^{it_2} \\right\\|^2_{2k} \\leq \\Lambda_k(A;u) + o_{T\\to\\infty}(1),\n\\]\nwhich is a straightforward consequence of our definition of $\\Lambda_k(A;u)$.\n\\end{proof}\n\nWe will use the following stability property of $\\Lambda$-constants which helps us to work with subsets.\n\n\\begin{Corollary} \\label{corr:stability}\nSuppose that $A \\subset \\mathbb Q$, that $u$ is a non-zero rational and $A' \\subset A$. Then\n\\[\n\\Lambda_k(A'; u) \\leq \\Lambda_k(A; u).\n\\]\nIn particular,\n\\[\n\\tilde E^{1\/k}_k(A';u) \\leq \\Lambda_k(A;u)|A'|.\n\\]\nand\n\\[\n\\tilde E_k(A;u) \\leq \\Lambda^k_k(A;u)|A|^k.\n\\]\n\\end{Corollary}\n\\begin{proof}\n The first claim follows from the observation that any set of weights $\\{w_a \\}_{a \\in A'}$ with $\\sum w^2_a = 1$ can be trivially extended to a set of weights $\\{ w_a \\}_{a \\in A}$ by assigning zero weight to the elements in $A \\setminus A'$. Next observe that $E_k$ is just $E_{k, w}$ with all the weights being one and apply Lemma~\\ref{lm:anyweights}.\n\\end{proof}\n\nThe next lemma records that any set with small separating factors also has a small $\\Lambda$-constant.\n\\begin{Lemma} \\label{lem:lambdasep}\nLet $A \\subset \\QQ$ be $\\psi$-separating. Then for any $u \\in \\QQ \\setminus \\{0\\}$\n\\[ \\Lambda_k(A;u) \\leq \\psi .\\]\n\\end{Lemma}\n\n\\begin{proof} Let $w$ be any set of weights on $A$ that satisfy \\eqref{eq:weightsnorm}. Write\n\\[A= \\bigcup_{a \\in A} a Y_a \\]\nwith $Y_a= \\{1\\}$ for all $a \\in A$. Then by the definition of $\\psi$-separating, it follows that for any non-zero $u \\in \\QQ$\n\\[ \\tilde E_{k,w}(A;u) ^{1\/k} \\leq  \\psi \\sum_{a \\in A} \\tilde E_{k,w} (\\{a \\};u)^{1\/k} = \\psi \\sum_{a \\in A} (w(a)^{2k})^{1\/k} = \\psi.\\]\n\\end{proof}\n\n\n\\begin{Lemma} \\label{lem:subset}\nLet $A \\subset \\mathbb Q$ be a finite set with $|AA| \\leq K|A|$ and let $u$ be a non-zero rational number. Suppose that $A' \\subset A$ and $A'$ is $\\psi$-separating. Then\n\\[ \\Lambda_k(A;u) \\leq K^4 \\left(\\frac{|A|}{|A'|-1}\\right)^2 \\psi. \\]\n\\end{Lemma}\n\n\\begin{proof}\n\nLet $w$ be an arbitrary set of weights on $A$ such that $\\sum_{a \\in A} w(a)^2 =1$. We seek a suitable upper bound for\n\\[ \\left\\|  \\sum_{a\\in A}w_aa^{it_1}(a+u)^{it_2} \\right \\|_{2k}^2 .\\]\nFor a fixed $z \\in A\/A'$, define a set of weights $w^{(z)}$ on $zA'$ by taking $w^{(z)}(za')=w(za')$ if $za' \\in A$ and $w^{(z)}(za')=0$ otherwise. Define \n\\[R_{(A\/A'),A'}(x):=|\\{(s,a) \\in (A\/A') \\times A' : sa=x \\}| \\]\nand note that $R_{(A\/A'),A'}(x) \\geq |A'|-1$ for all $x \\in A$. This is because, for all non-zero $a' \\in A'$, $x=(\\frac{x}{a'})a'$. Therefore,\n\\begin{align*} \\left\\| \\sum_{z \\in A\/A'} \\sum_{a' \\in A'} w^{(z)}(za')(za')^{it_1}(za'+u)^{it_2} \\right \\|_{2k}\n&= \\left\\| \\sum_{a \\in A}R_{(A\/A'),A'} (a) w(a)a^{it_1}(a+u)^{it_2} \\right \\|_{2k}\n\\\\& \\geq |A'| \\left\\|  \\sum_{a\\in A}w_aa^{it_1}(a+u)^{it_2} \\right \\|_{2k}.\n\\end{align*}\nOn the other hand, by the triangle inequality and Lemma \\ref{lm:anyweights}\n\\begin{align*}\n\\left\\| \\sum_{z \\in A\/A'} \\sum_{a' \\in A'} w^{(z)}(za')(za')^{it_1}(za'+u)^{it_2} \\right \\|_{2k}\n& \\leq \\sum_{z \\in A\/A'} \\left\\|  \\sum_{a' \\in A'} w^{(z)}(za')(za')^{it_1}(za'+u)^{it_2} \\right \\|_{2k}\n\\\\& \\leq  \\sum_{z \\in A\/A'} \\Lambda_k(zA'; u)^{1\/2} + o_{T \\rightarrow \\infty}(1).\n\\end{align*}\nSince $A'$ is $\\psi$-separating, it follows from Lemma \\ref{lem:lambdasep} that $\\Lambda_k(zA';u) = \\Lambda_k(A';u\/z) \\leq \\psi $. We also have\n\\[|A\/A'| \\leq |A\/A| \\leq \\frac{|AA|^2}{|A|} \\leq K^2 |A|, \\]\nby the Ruzsa Triangle Inequality (see \\cite{TV}). It therefore follows that\n\\[ \\left\\|  \\sum_{a\\in A}w_aa^{it_1}(a+u)^{it_2} \\right \\|_{2k} \\leq K^2 \\left( \\frac{|A|}{|A'|-1} \\right ) \\psi^{1\/2} + o_{T \\rightarrow \\infty} (1),\n\\]\nand the result follows.\n\\end{proof}\n\nCombining this with Theorem \\ref{thm:goodsubset} gives the following, which is our main result concerning $\\Lambda$-constants.\n\n\\begin{Theorem} \\label{thm:lambda}\nGiven $0<\\gamma < 1\/2$, there exists a positive constants $C=C(\\gamma, k)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$ and any non-zero rational $u$,\n\\[ \\Lambda_k(A;u) \\leq K^C|A|^{\\gamma} .\\]\n\\end{Theorem}\n\n\\section{Concluding the proofs} \\label{sec:conclusion}\n\nIn this section we conclude the proof of Theorem \\ref{thm:mainmain}, which is the main theorem of this paper, and Theorem~\\ref{thm:BS_almost_subgroups} announced in the introduction. Both theorems are restated below for the convenience of the reader.\n\n\\begin{Theorem} \\label{thm:mainmainagain}\nFor all $b \\in \\mathbb Z$, there exists $k=k(b)$ such that for any finite set $A \\subset \\mathbb Q$ and any non-zero rational $u$,\n\\[  \\max \\{ |A^{(k)}|, |(A+u)^{(k)}| \\} \\geq |A|^b  \\]\n\\end{Theorem}\n\n\\begin{proof} Fix $b$ and assume that\n\\[|A^{(k)}| < |A|^b \\]\nfor some sufficiently large $k=2^l$. The value of $l$ (and thus also that of $k$) will be specified at the end of the proof.\nSince $|A^{(2^l)}| < |A|^b$, it follows that\n\\[ \\frac{|A^{(2^l)}|}{|A^{(2^{l-1})}|} \\frac{|A^{(2^{l-1})}|}{|A^{(2^{l-2})}|} \\cdots \\frac{|A^{(2)}|}{|A|} < |A|^{b-1}\\]\nand thus there is some integer $l_0 \\leq l$ such that\n\\[ \\frac{|A^{(2^{l_0+1})}|}{|A^{(2^{l_0})}|} < |A|^{\\frac{b-1}{l}} .\\]\nTherefore, writing $k_0=2^{l_0}$ and $B=A^{(k_0)}$, we have\n\\[|BB| < |B| |A|^{\\frac{b-1}{l}}. \\]\nAlso, for any non-zero $\\lambda \\in \\mathbb Q$, $|(\\lambda B)(\\lambda B)| <|B||A|^{\\frac{b-1}{l}}$. Therefore, by Theorem \\ref{thm:lambda},\n\\[ \\Lambda_h(\\lambda B;u) \\leq |A|^{C\\frac{b-1}{l}} |B|^{\\gamma} \\leq |A|^{C\\frac{b-1}{l}+ \\gamma b}  \\]\nwhere $C=C(h, \\gamma)$ and $h, \\gamma$ will be specified later.\n\nNow, for some $\\lambda \\in \\mathbb Q$, we have $A \\subset \\lambda B$, and thus by Corollary \\ref{corr:stability} and Lemma \\ref{lem:CSbasic}\n\\[ \\frac{|A|^2}{\\max \\{|A^{(h)}|,|(A+u)^{(h)}| \\}^{2\/h}} \\leq \\tilde E_h^{1\/h}(A;u) \\leq |A| \\Lambda_h(\\lambda B;u) \\leq |A|^{1+C\\frac{b-1}{l}+ \\gamma b} .\\]\nThis rearranges to\n\\[ \\max \\{|A^{(h)}|,|(A+u)^{(h)}| \\} \\geq |A|^{\\frac{h}{2}(1-C\\frac{b-1}{l}- \\gamma b)} .\\]\nChoose $\\gamma=1\/100b$ and $h=4b$. Then $C=C(h,\\gamma)=C(b)$ and we have\n\\[ \\max \\{|A^{(h)}|,|(A+u)^{(h)}| \\} \\geq |A|^{\\frac{h}{2}(99\/100-C(b)\\frac{b-1}{l})} .\\]\nThen choose $l=(b-1)4C$ to get\n\\[ \\max \\{|A^{(h)}|,|(A+u)^{(h)}| \\} \\geq |A|^{\\frac{h}{4}}=|A|^b .\\]\nNote that the choice of $l$ depends only on $b$ and thus $k=2^{4C(b-1)}=k(b)$.\nIn particular, since $k > h$, we conclude that\n\\[ \\max \\{|A^{(k)}|,|(A+u)^{(k)}| \\} \\geq |A|^b ,\\]\nas required. \n\n\\end{proof}\n\nTheorem \\ref{thm:lambda} also implies Theorem \\ref{thm:main1}. The statement is repeated below for the convenience of the reader.\n\n\\begin{Theorem}\nGiven $0<\\gamma < 1\/2$ and any integer $k \\geq 2$, there exists a positive constant $C=C(\\gamma, k)$ such that for any finite $A \\subset \\mathbb Q$ with $|AA|=K|A|$ and any non-zero rational $u$,\n\\[ | (A+u) ^{(k)}|  \\geq \\frac{|A|^{k(1-\\gamma)-1}}{K^{Ck}}.\\]\n\\end{Theorem}\n\n\\begin{proof} Define $w(a)= 1 \/ |A|^{1\/2}$ for all $a \\in A$ and note that \\eqref{eq:weightsnorm} is satisfied. Furthermore, for this set of weights $w$,\n\\begin{equation}\n\\tilde E_{k,w}(A;u) = \\frac{\\tilde E_k(A;u)}{|A|^{k}} \\geq \\frac{|A|^{k}}{|A^{(k)}||(A+u)^{(k)}|},\n\\label{eq:trivialweights}\n\\end{equation}\nwhere the inequality comes from Lemma \\ref{lem:CSbasic}. It follows from Theorem \\ref{thm:lambda} that there exists a constant $C=C(\\gamma,k)$ such that for any $u \\in \\mathbb Q \\setminus \\{0\\}$, $\\Lambda_{k}(A;u) \\leq K^C|A|^{\\gamma}$. Consequently, by the definition of $\\Lambda_{k}(A;u)$,\n\\[ \\tilde E_{k,w}(A;u) \\leq K^{Ck}|A|^{\\gamma k}. \\]\nCombining this with \\eqref{eq:trivialweights}, it follows that\n\\begin{equation}\n|A^{(k)}||(A+u)^{(k)}| \\geq \\frac{|A|^{k(1-\\gamma)}}{K^{Ck}}.\n\\label{eq:energybound}\n\\end{equation}\nFinally, since $|AA| \\leq K|A|$, it follows from the Pl\\\"{u}nnecke-Ruzsa Theorem that $|A^{(k)}| \\leq K^k|A|$. Inserting this into \\eqref{eq:energybound} completes the proof.\n\n\n\n\n\n\\end{proof}\n\n\nWe now turn to the proof of Theorem~\\ref{thm:BS_almost_subgroups}. Recall its statement.\n\n\\begin{Theorem} For any $\\gamma > 0$ there is $C(\\gamma) > 0$ such that for any $K$-almost subgroup $A \\subset \\mathbb{Q}$ and fixed non-zero $c_1, c_2 \\in \\mathbb{Q}$ the number $A(2, K)$ of solutions $(x_1, x_2) \\in A^2$ to \n $$\n c_1x_1 + c_2 x_2 = 1\n $$\n is bounded by \n $$\n A(2, K) \\leq |A|^\\gamma K^C.\n $$\n\n\\end{Theorem}\n\\begin{proof}\n\tLet $S \\subset A$ be the set of $x_1 \\in A$ such that $c_1x_1 + c_2x_2 = 1$ for some $x_2 \\in A$. Since the projection $(x_1, x_2) \\to x_1$ is injective, it suffices to bound the size of $S$.\n    \n    Since $S \\subset A$, by Theorem~\\ref{thm:lambda} and\n    Corollary~\\ref{corr:stability} for any non-zero $u$ \n$$\n\\tilde E_k(S; u) \\leq K^{k C(\\gamma', k)}|A|^{k\\gamma'} |S|^k\n$$\nwith the parameters $0 < \\gamma' < 1\/2, k \\geq 2$ to be taken in due course.\n\nIn particular, by Lemma~\\ref{lem:CSbasic}\n$$\n|S|^k \\leq \\left(K^{k C(\\gamma', k)}|A|^{k\\gamma'} |S|^k \\right)^{1\/2} \\max \\{ |S^k|, |(S-1\/c_1)^k| \\}.\n$$\n\nOn the other hand, $S \\subseteq A$ and $(S - 1\/c_1) \\subseteq (c_2\/c_1)A$, so by the Pl\\\"{u}nnecke-Ruzsa inequality\n$$\n\\max \\{ |S^k|, |(S-1\/c_1)^k| \\} \\leq |A^{(k)}| \\leq K^k|A|.\n$$\n\nWe then have\n$$\n|S| \\leq |A|^{\\gamma' + 2\/k} K^{C + 2},\n$$\nand taking $k = \\lfloor 2\/\\gamma' \\rfloor + 1$ and $\\gamma'= \\gamma\/2$, the claim follows.\n\n\\end{proof}\n\n\\section{Graph Fibering} \\label{sec:fibers}\nSuppose $Z_1$ and $Z_2$ abelian groups, with finite subsets $A,B\\subset Z_1\\times Z_2$. We will write $z_1\\oplus z_2$ for an element of $Z_1\\times Z_2$. We will write, for $x\\in X\\subset Z_1\\times Z_2$ with $\\pi_1(x)=x_1$, \\[X_2(x_1)=\\{x_2\\in\\pi_2(X):x_1\\oplus x_2\\in X\\}.\\] Suppose $G\\subset A\\times B$. Denote by $\\pi_1$ and $\\pi_2$ the projections onto the first and second coordinates of $Z_1\\times Z_2$ respectively. The set $G$ is interpreted as a bipartite graph on $A$ and $B$, and it can be decomposed into a union by considering the fibers of $\\pi_1$. Indeed, let \\[G_1=\\{(\\pi_1(a),\\pi_1(b)):(a,b)\\in G\\}\\] and for $(a_1,b_1)\\in G_1$, let \\[G_2(a_1,b_1)=\\{(a_2,b_2):(a_1\\oplus a_2,b_1\\oplus b_2)\\in G\\}\\subset \\pi_2(A)\\times\\pi_2(B).\\]\nRecall the notation \\[A+_G B=\\{a+b:(a,b)\\in G\\}.\\]\n\nOne of the primary reasons for decomposing a graph this way is that it behaves nicely with addition along the graph.\n\\begin{Lemma}\\label{FiberGraphSum}\nSuppose $A$ and $B$ are finite subsets of $Z_1\\times Z_2$. Then for $G\\subset A\\times B$ we have\n\\[|A+_G B|\\geq |\\pi_1(A)+_{G_1}\\pi_1(B)|\\min_{(a_1,b_1)\\in \\pi_1(A)\\times \\pi_1(B)}|A_2(a_1)+_{G_2(a_1,b_1)}B_2(b_1)|.\\]\n\\end{Lemma}\n\\begin{proof}\nWrite \n\\[A+_G B\\supseteq \\bigcup_{s\\in \\pi_1(A+_G B)} \\bigcup_{\\substack{(a_1\\oplus a_2,b_1\\oplus b_2)\\in G\\\\ a_1+b_1=s}}\\{(s\\oplus (a_2+b_2))\\}.\\]\nNext, from the observation that the first union above is disjoint, and the fact that $\\pi_1(A+_G B)=\\pi_1(A)+_{G_1}\\pi_1(B)$, we have\n\\[|A+_G B|\\geq\\sum_{s\\in \\pi_1(A)+_{G_1}\\pi_1(B)}\\big|\\bigcup_{\\substack{(a_1\\oplus a_2,b_1\\oplus b_2))\\in G\\\\ a_1+b_1=s}}\\{(s\\oplus (a_2+b_2))\\}\\big|.\\]\nSince, for fixed $a_1,b_1$,\n\\[\\bigcup_{\\substack{(a_1\\oplus a_2,b_1\\oplus b_2)\\in G\\\\ a_1+b_1=s}}\\{a_2+b_2\\}\\supseteq A_2(a_1)+_{G_2(a_1,b_1)} B_2(b_1)\\]\nthe lemma follows.\n\\end{proof}\n\n\\begin{Lemma}[Regularized decomposition]\\label{lm:FiberingLemma}\nLet $Z_1$ and $Z_2$ be abelian groups and let $A,B\\subset Z_1\\times Z_2$ be finite sets. Suppose that $\\delta>0$, $K\\geq 1$ and $G\\subset A\\times B$ are such that \\[|G|\\geq \\delta |A||B|,\\] and\n\\[|A+_G B|\\leq K(|A||B|)^{1\/2}.\\] There are absolute constants $c,C>0$, subsets $A'\\subset A$ and $B'\\subset B$, and a subset $G'\\subset A'\\times B'$ with the following properties.\n\\begin{enumerate}\n\\item(Uniform fibers) If \n\\ben \\label{eq:M_1M_2definition}\n\t\tM_A=|\\pi_1(A)|,\\ M_B=|\\pi_1(B)|\n\\een \nthen there are numbers $m_A$ and $m_B$ satisfying \n\\ben \n\tM_Am_A&\\geq c\\delta^2(\\log(K\/\\delta))^{-1}|A|,\\label{eq:FiberingLemmaSetSizes}\\\\ \n\tM_Bm_B&\\geq c\\delta^2(\\log(K\/\\delta))^{-1}|B|,\\label{eq:FiberingLemmaSetSizes2}\\\\\n\tm_A,m_B&\\geq c\\delta^{10}K^{-4}\\max_{a_1\\in\\pi_1(A),b_1\\in\\pi_1(B_1)}\t(|A_2(a_1)|+|B_2(b_1)|),\n\\een\nand such that we have approximately uniform fibers: \t\t\\ben\\label{eq:FiberingLemmaFiberSizes}\n|(A')_2(a_1)|\\approx m_A,\\ |(B')_2(b_1)|\\approx m_B\n\\een \nfor $a_1\\in \\pi_1(A')$ and $b_1\\in \\pi_1(B')$.\n\\item(Uniform graph fibering) For some $\\delta_1,\\delta_2>0$ satisfying\n\\ben \\label{eq:FiberingLemmaDeltaSizes}\n\t\\delta_1\\delta_2>c(\\log (K\/\\delta))^{-3}\\delta\n\\een we have that the first coordinate subgraph is dense: \n\\ben \\label{eq:FiberingLemmaBaseGraphSize}\n\t|G_1'|\\geq \\delta_1M_AM_B,\n\\een and that the subgraph has dense fibers: for each $(a_1,b_1)\\in G'_1$ we have\n\\ben \\label{eq:FiberingLemmaFiberGraphSize}\n\t|G'_2(a_1,b_1)|\\geq \\delta_2m_Am_B.\n\\een\n\\item(Bounded doubling) For some $K_1,K_2>0$ with\n\\ben\\label{eq:FiberingLemmaDoublingConstants}\n\tK_1K_2\\leq C\\delta^{-2}(\\log K)K\n\\een\nwe have \n\\ben\\label{eq:FiberingLemmaK1}\n\t|\\pi_1(A')+_{G'_1}\\pi_1(B')|= K_1(M_AM_B)^{1\/2},\n\\een\nand for each $(a_1,b_1)\\in G_1'$,\n\\ben \\label{eq:FiberingLemmaK2}\n\t|\\pi_2(A')+_{G'_2(a_1,b_1)}\\pi_2(B')|\\approx K_2(m_Am_B)^{1\/2}.\n\\een\n\\end{enumerate}\n\\end{Lemma}\n\\subsection{Proof of Theorem \\ref{lm:FiberingLemma}}\nWe will produce the sets $A'$ and $B'$ after a sequence of refinements. One such refinement comes from the following lemma. Here, and in what follows, when $G\\subseteq A\\times B$ we write $\\deg_G a$ (respectively, $\\deg_G b$) for the size of $\\{b'\\in B:(a,b')\\in G\\}$ (respectively, the size of $\\{a'\\in A:(a',b)\\in G\\}$).\n\\begin{Lemma} \\label{lemma:Step0}\nLet $A$ and $B$ be finite sets and $G\\subseteq A \\times B$ of size $\\delta |A||B|$. Then there exist $A' \\subset A, B' \\subset B$ and $G' \\subset G \\cap (A' \\times B')$ such that \n\\begin{itemize}\n\\item $\\deg_{G'} a\\geq \\frac{\\delta}{4} |B|$,\n\\item $\\deg_{G'} b\\geq \\frac{\\delta}{4} |A|$,\n\\item $|A'| \\geq \\frac{\\delta}{2}|A|$,\n\\item $|B'| \\geq \\frac{\\delta}{2}|B|$, and \n\\item $|G'| \\geq \\frac{\\delta}{2} |A||B|$\n\\end{itemize}\nfor any $a \\in A', b \\in B'$.\n\\end{Lemma}\n\\begin{proof}\nRemove from $A$ (respectively, $B$) one by one all vertices with degree less than $\\delta|A|\/4$ (respectively, $\\delta|B|\/4$), until both $A$ and $B$ contain only vertices of degree at least $\\delta|A|\/4$ (respectively, $\\delta|B|\/4$) in the remaining graph. At the end of this process, we cannot have removed more than $\\delta|A||B|\/2$ edges. Indeed, we remove at any stage at most $\\delta|B|\/4$ edges adjacent to a vertex in $|A|$ (and we can remove at most $|A|$ such vertices) or else at most $\\delta|A|\/4$ edges adjacent to a vertex in $B$ (and we can remove at most $|B|$ such vertices). Take $A'$ and $B'$ to be the sets of survived vertices in $A$ and $B$ respectively and $G' := G \\cap (A' \\times B')$.\n\\end{proof}\n\nNow, set $|A|=N_A$ and $|B|=N_B$. In view of the above lemma, and passing to subsets if necessary, we may assume \\[|A|\\geq \\frac{1}{2}\\delta N_A,\\ |B|\\geq \\frac{1}{2}\\delta N_B,\\] \\[|G|\\geq \\frac{1}{2}\\delta N_AN_B\\] and that for any $a\\in A$ and $b\\in B$ we have\n\\[\\deg_G a\\geq \\frac{1}{4}\\delta|B|,\\ \\deg_G b\\geq \\frac{1}{4}\\delta|A|.\\]\nFirst, we may assume without loss of generality that \\[n_A=\\max_{a_1\\in\\pi_1(A)}|A_2(a_1)|\\geq\\max_{b_1\\in\\pi_1(B)}|B_2(b_1)|\\]\nIt is also useful to observe that, if $a\\in A$ then $|\\{a\\}+_G B|=\\deg_G a$, where $\\deg_G a$ is the number of neighbours of $a$ in $G$. So, \n\\[\\delta N_B\\leq \\frac{1}{N_A}\\sum_{a\\in A}\\deg_G a\\leq |A+_G B|\\leq K(N_AN_B)^{1\/2}.\\]\nWe can apply the same argument, reversing the roles of $A$ and $B$, and we have proved\n\\begin{equation}\\label{SizeComparison}\n\\delta N_B^{1\/2}\\leq KN_A^{1\/2},\\ \\delta N_A^{1\/2}\\leq KN_B^{1\/2}.\n\\end{equation}\nHaving assumed this, our first order of business is to establish property (1) for $B'$.\n\n\\subsubsection{Regularization of $B$}\n\nLet $a_1\\in\\pi_1(A)$ be such that $|A_2(a_1)|=n_A$. Then $a_1\\oplus A_2(a_1)$ consists of $n_A$ elements of $A$ each with at least $\\frac{1}{4}\\delta|B|$ neighbours in $B$. Thus\n\\[|(a_1\\oplus A_2(a_1)\\times B)\\cap G|\\geq \\frac{1}{4}\\delta n_A|B|.\\] Let\n\\[B'=\\left\\{b\\in B:|\\{a_2:(a_1\\oplus a_2,b)\\in G\\}|\\geq \\frac{1}{8}\\delta n_A\\right\\}\\]\n\\begin{equation}\\label{Fibre1}\n|B'|\\geq \\frac{1}{8}\\delta |B|\\geq\\frac{1}{16}\\delta^2 N_B\n\\end{equation} and such that for each $b\\in B'$ we have\n\\[|((a_1\\oplus A_2(a_1))\\times\\{b\\})\\cap G|\\geq \\frac{1}{8}\\delta n_A.\\]\nMoreover, since every element in $B$ has at least $\\frac{1}{4}\\delta N_A$ neighbours in $A$, we have\n\\[|(A\\times B')\\cap G|\\geq\\frac{1}{4}\\delta N_A|B'|.\\] If $k=|\\pi_1(B')|$ then there are elements $b_1\\oplus b_1',\\ldots,b_k\\oplus b_k'$ with the $b_i$ distinct, and for each of them the sets\n\\[a_1\\oplus A_2(a_1)+b_i\\oplus b_i'\\] are disjoint, since their first coordinates are $a_1+b_i$ and are distinct. Each of these sets contains at least $\\frac{1}{8}\\delta n_A$ distinct elements of $A+_G B$ since each element of $B'$ has that many neighbours in $G$. From this it follows that\n\\[|a_1\\oplus A_2(a_1)+_G B'|\\geq \\frac{1}{8}\\delta n_A|\\pi_1(B')|\\] and so\n\\[\\frac{1}{8}\\delta n_A|\\pi_1(B')|\\leq |A+_G B|\\leq K(N_AN_B)^{1\/2}\\leq \\frac{K^2}{\\delta}N_B.\\] Here we have used the inequality (\\ref{SizeComparison}). Next, we define\n\\begin{equation}\\label{Fibre2}\nB''=\\bigcup_{\\substack{1\\leq i\\leq k\\\\ |B'_2(b_i)|\\geq 10^{-4}\\delta^5K^{-2}n_A}}b_i\\oplus B'_2(b_1).\n\\end{equation}\nBy (\\ref{Fibre2}) and (\\ref{Fibre1}),\n\\[|B'\\setminus B''|\\leq |\\pi_1(B')|10^{-4}\\delta^5K^{-2}n_A\\leq 10^{-3}\\delta^3N_B\\leq \\frac{\\delta}{10}|B'|.\\]\nNow, we have already assumed that $\\max_{b_1\\in\\pi_1(B)}|B_2(b_1)|\\leq n_A$, so applying a dyadic partition to the range $10^{-4}\\delta^5K^{-2}n_A\\leq m\\leq n_A$, we find a value of $m_B$ in this range and a subset \n\\[B'''=\\bigcup_{\\substack{b_1\\in\\pi_1(B')\\\\m_B\\leq |B''_2(b_1)|\\leq 2m_B}}b_1\\oplus B'_2(b_1)\\] which has size $|B'''|\\gg\\log(K\/\\delta)^{-1}|B''|$.\nThus\n\\[|B'''|\\gg\\frac{|B''|}{\\log(K\/\\delta)}\\gg \\frac{|B'|}{\\log(K\/\\delta)}\\gg \\frac{\\delta^2}{\\log(K\/\\delta)}N_B.\\]\nSince each element of $B$ has at $\\frac{1}{8}\\delta N_A$ neighbours in $G$, we further have\n\\[|(A\\times B''')\\cap G|\\geq \\frac{1}{8}\\delta N_A|B'''|.\\]\nIf $M_B=|\\pi_1(B''')|$, then because each element of $\\pi_1(B''')$ has about $m_B$ fibers, we have\n\\[|B'''|\\approx m_BM_B.\\]\nRedefine $B'=B'''$ and $N_B'=|B'|$. Then we have shown that \n\\[N_B'\\gg \\frac{\\delta^2}{\\log(K\/\\delta)}N_B.\\]\n\n\\subsubsection{Regularization of $A$}\n\nLet \\[A'=\\bigcup_{\\substack{a_1\\in\\pi_1(A)\\\\|A_2(a_1)|\\geq 10^{-5}\\delta^3K^{-2}m_B}}a_1\\oplus A_2(a_1).\\] \nWe first estimate $|A\\setminus A'|$. We write $A''=A\\setminus A'$, so that for each $a\\in A''$ we have \n\\begin{equation}\\label{A''}\n|A_2(a_1)|<10^{-5}\\delta^3K^{-2}m_B\n.\\end{equation} We will show $|(A''\\times B')\\cap G|\\leq \\frac{\\delta}{40} N_AN_B'$. To see why, assume the contrary. Then there is a $b_1\\in \\pi_1(B')$ with \\[|(A''\\times b_1\\oplus B'_2(b_1))\\cap G|\\geq\\frac{\\delta}{100}N_Am_B.\\] Indeed, each of the vertex sets $b_1\\oplus B'_2(b_1)$ are disjoint and have size $m_B$ up to a factor of $2$. Now let $A'''\\subset A''$ be the set of those $a$ for which\n\\[|(\\{a\\}\\times b_1\\oplus B'_2(b_1))\\cap G|\\geq \\frac{\\delta}{200}m_B.\\] From the definition, it follows that\n\\begin{equation}\\label{A'''}\n|A'''|\\geq \\frac{\\delta}{200}N_A.\n\\end{equation}\nLet \\[M=\\max_{a_1\\in\\pi_1(A''')}|A'''_2(a_1)|.\\]\nWe have\n\\[|A'''+_G(b_1\\oplus B'_2(b_1)))|\\leq |A+_G B|\\leq K(N_AN_B)^{1\/2}\\leq \\frac{K^2}{\\delta}N_A.\\] Because every element of $A'''$ has at least $(\\delta\/200)m_B$ neighbours in $b_1\\oplus B'_2(b_1)$, and because for each $a_1\\in \\pi_1(A''')$ the sets $(a_1\\oplus A_2'''(a_1))+_G(b_1\\oplus B'_2(b_1))$ are disjoint, we get\n\\[|A'''+_G(b_1\\oplus B'_2(b_1)))|\\geq (\\delta\/200) m_B|\\pi_1(A''')|.\\]\nIn view of (\\ref{A'''}) \\[|\\pi_1(A''')|\\geq \\frac{|A'''|}{M}\\geq \\frac{\\delta}{200M}N_A,\\] we obtain the bound\n\\[\\frac{\\delta^2}{4\\cdot 10^4M}N_Am_B\\leq \\frac{K^2}{\\delta}N_A\\] whence \n\\[M>\\frac{\\delta^3m_B}{10^5 K^2},\\] which contradicts (\\ref{A''}) and the definition of $M$.\nBy what we have just shown,\n\\[|(A'\\times B')\\cap G|\\geq\\frac{\\delta}{8}N_AN_B'.\\] Now, for each $a\\in A'$, we certainly have\n\\[|A_2(a_1)|\\leq n_A\\leq 10^4m_B \\delta^{-5}K^2\\] the final estimate coming from the bounds on the range range of $m_B$. Thus we partition the range \n\\[10^{-5}\\delta^3K^{-2}m_B\\leq |A_2(a_1)|\\leq 10^4m_B \\delta^{-5}K^2\\] dyadically, to find an $m_A$ in this range such that\n\\[A''''=\\bigcup_{\\substack{a_1\\in\\pi_1(A)\\\\m_A\\leq |A_2(a_1)|\\leq 2m_A}}a_1\\oplus A_2(a_1)\\]\nsatisfies\n\\[|(A''''\\times B')\\cap G|\\gg\\frac{\\delta}{\\log(K\/\\delta)}N_AN_B'.\\] Moreover, since $|(A''''\\times B')\\cap G|\\leq |A''''|N_B'$ we have $|A''''|\\gg \\delta(\\log(K\/\\delta))^{-1}N_A$. If we define $M_A=|\\pi_1(A'''')|$ then we have\n\\[|A''''|\\approx M_Am_A\\] as needed. We relabel $A'=A''''$ and $N_A'=|A'|$, observing that\n\\[N_A'\\gg \\frac{\\delta}{\\log(K\/\\delta)}N_A\\] and we are ready to proceed to the next step.\n\n\\subsubsection{Regularizing the graph fibers}\n\nSo far we have found subsets $A'$ and $B'$, and an absolute constant $c>0$, satisfying \n\\[|(A'\\times B')\\cap G|\\geq c\\frac{\\delta}{\\log(K\/\\delta)}|A'||B'|,\\]\n\\[|A'|\\approx m_AM_A\\geq c\\frac{\\delta}{\\log(K\/\\delta)}N_A,\\]\nand\n\\[|B'|\\approx m_BM_B\\geq c\\frac{\\delta^2}{\\log(K\/\\delta)}N_B.\\]\nFurthermore, each of $A'$ and $B'$ have fibers above $\\pi_1$ of size roughly $m_A$ and $m_B$ respectively. Recall that for $(a_1,b_1)\\in\\pi_1(A')\\times \\pi_1(B')$ we have the graph\n\\[G_2(a_1,b_1)=\\{(a_2,b_2)\\in A'_2(a_1)\\times B'_2(b_1):(a_1\\oplus a_2,b_1\\oplus b_2)\\in G\\}.\\] Because we have regularized the fibers of $A'$ and $B'$, each of these graphs has cardinality obeying \\[|G_2(a_1,b_1)|\\leq 4m_Am_B.\\] By a slight abuse of notation, we let\n\\[G_1=\\{(\\pi_1(a),\\pi_1(B)):(a,b)\\in (A'\\times B')\\cap G\\}\\] and define\n\\[G_1'=\\left\\{(a_1,b_1)\\in \\pi_1(A')\\times \\pi_1(B'):|G_2(a_1,b_1)|\\geq \\frac{c\\delta}{16\\log(K\/\\delta)}m_Am_B\\right\\}.\\]\nSince\n\\[\\sum_{(a_1,b_1)\\in\\pi_1(A')\\times\\pi_1(B')}|G_2(a_1,b_1)|=|(A'\\times B')\\cap G|\\geq c\\frac{\\delta}{\\log(K\/\\delta)}|A'||B'|\\]\nit follows that\n\\[\\sum_{(a_1,b_1)\\in G_1'}|G_2(a_1,b_1)|\\geq c\\frac{\\delta}{2\\log(K\/\\delta)}|A'||B'|.\\]\nBy a dyadic pigeon-holing for $\\delta'$ in the range $c\\delta(\\log(K\/\\delta))^{-1}\\leq \\delta '\\leq 4$, we can find $\\delta'\\gg \\delta (\\log(K\/\\delta))^{-1}$ such that\n\\[G_1''=\\left\\{(a_1,b_1)\\in G_1':\\delta'm_Am_B\\leq|G_2(a_1,b_1)|\\leq2\\delta'm_Am_B\\right\\}\\]\ncertainly satisfies\n\\[\\sum_{(a_1,b_1)\\in G_1''}|G_2(a_1,b_1)|\\gg c\\frac{\\delta}{(\\log(K\/\\delta))^2}|A'||B'|.\\]\nFrom this estimate, it also follows that\n\\[|G_1''|\\gg \\frac{\\delta}{\\delta'(\\log(K\/\\delta))^2}M_AM_B.\\]\nLet us relabel $G_1''$ as $G_1'$ and set \n\\[G'=\\{(a,b)\\in A'\\times B':(\\pi_1(a),\\pi_1(b))\\in G_1'\\}.\\] We move on to the final step of the lemma.\n\n\\subsubsection{Regularizing the doubling constant}\n\nFor $(a_1,b_1)\\in \\pi_1(A')\\times\\pi_1(B')$ we define\n\\[K_+(G_2(a_1,b_1))=\\frac{|A'_2(a_1)+_{G_2(a_1,b_1)}B'_2(b_1)|}{(|A'_2(a_1)||B'_2(b_1)|)^{1\/2}}.\\]\nThis quantity measure the growth of sumsets on the fibres lying above a pair $(a_1,b_1)$. Now define\n\\[H=\\{(a_1,b_1)\\in G_1':K_+(G_2(a_1,b_1))>C(\\log(K\/\\delta))^3\\delta^{-10}K\\}.\\]\nProvided $C$ is large enough we have $H\\leq \\frac{1}{10}|G_1'|$. To see this, first observe the trivial bound\n\\begin{equation}\\label{HSumset}|\\pi_1(A')+_H\\pi_1(B)|\\geq\\frac{|H|}{\\min\\{|\\pi_1(A')|,|\\pi_1(B')|\\}}\\geq \\frac{|H|}{(M_AM_B)^{1\/2}}.\n\\end{equation}\nLet \\[G_H=\\{(a_1,a_2)\\in G:(\\pi_1(a_1),\\pi_1(a_2))\\in H\\}\\subset G.\\]\nAlso, for $(a_1,b_1)\\in H$ we have\n\\[(G_H)_2(a_1,b_1)=G_2(a_1,b_1)\\] so that by Lemma \\ref{FiberGraphSum}\n\\[|A'+_G B'|\\geq |\\pi_1(A')+_H\\pi_1(B')|\\min_{(a_1,b_1)\\in H}(|A'_2(a_1)+_{G_2(a_1,b_1)}B'_2(b_1)|).\\]\nBy the definition of $H$ and (\\ref{HSumset}) we see\n\\[K(N_AN_B)^{1\/2}\\geq |A'+_G B'|\\geq C\\frac{|H|}{(M_AM_B)^{1\/2}}(\\log(K\/\\delta))^3\\delta^{-10}K(m_Am_B)^{1\/2}.\\]\nUsing our estimates for $m_AM_A$, $m_BM_B$ and $G_1'$, the right hand side is \n\\[C\\frac{|H|}{M_AM_B}(\\log(K\/\\delta))^3\\delta^{-10}K(M_Am_AM_Bm_B)^{1\/2}\\geq cCK(N_AN_B)^{1\/2}\\frac{|H|}{|G_1'|}.\\] Thus for $C$ sufficiently large in terms of $c$ (which was absolute), we have $|H|\\leq \\frac{1}{10}|G_1'|$. \nNow let $G_1''=G_1'\\setminus H$. We perform yet another dyadic pigeon-holing to find $K'\\leq C(\\log(K\/\\delta))^3\\delta^{-10}K$ such that\n\\[G_1'''=\\{(a_1,b_1)\\in G_1'':K'\\leq K_+(G_2(a_1,b_1))\\leq 2K'\\}\\] has cardinality \\[|G_1'''|\\gg \\frac{|G_1'|}{\\log(K\/\\delta)}.\\]\nNow, by Lemma \\ref{FiberGraphSum} along the subgraph of $G$ with first projection equal to $G_1'''$ we have\n\\[K(N_AN_B)^{1\/2}\\geq |\\pi_1(A')+_{G_1'''}\\pi_1(B')|K'(m_Am_B)^{1\/2}=K_+(G_1''')K'(M_Am_AM_Bm_B)^{1\/2},\\]\nwhere $K_+(G_1''')=|\\pi_1(A')+_{G_1'''}\\pi_1(A_2')|(M_AM_B)^{-1\/2}$. By the established bounds on $m_AM_A$ and $m_BM_B$, we get\n\\[K(N_AN_B)^{1\/2}\\gg K_+(G_1''')K'\\delta^{3\/2}\\log(K\/\\delta)(N_AN_B)^{1\/2}.\\]\nFrom this we see\n\\[K_+(G_1''')K'\\ll K\\log (K\/\\delta)\\delta^{3\/2}\\ll K\\log (K)\\delta^{2}.\\]\nNow let $G'=\\{(a,b)\\in A'\\times B':(\\pi_1(a),\\pi_1(b))\\in G_1'''\\}.$ Define $K_1=K_+(G_1''')$ and $K_2=K'$. Let $\\delta_2=\\delta'$ and $\\delta_1=c\\delta(\\delta_2(\\log(K\/\\delta))^3)^{-1}$. One then verifies that with these parameters, the claims of the lemma have all been justified.\n\n\n\\section{Iteration scheme} \\label{sec:iterationscheme}\n\nIn this section we will use Lemma \\ref{lm:FiberingLemma} in order to setup an iteration scheme. At each step we have a pair of sets $(\\mathcal{A}, \\mathcal{B})$ which correspond to a pair of additive sets $(A, B) := (\\mathcal{P}(\\mathcal{A}), \\mathcal{P}(\\mathcal{B}))$ and a graph $G$ on $A \\times B$, together with the data $(N, \\delta, K)$ such that:\n\\begin{enumerate}\n \t\\item $|A||B| = N \\label{eq:setupNGKdelta1}$\n\t\\item $|A +_G B| \\leq KN^{1\/2}$ \\label{eq:setupNGKdelta2} \n\t\\item $|G| \\geq \\delta N$ \\label{eq:setupNGKdelta3}.\n\\end{enumerate}\n\n\n\nApart from that, the setup above is equipped with a pair of functions $\\psi(N, \\delta, K)$, $\\phi(N, \\delta, K)$ (which are called \\emph{admissible} in \\cite{BC}). These functions are technical aids to carry out an induction type argument.\n\n\\begin{Definition}[Admissible pair of functions] \\label{def:admissible_pairs}\nA pair of functions $\\psi(N, \\delta, K)$, and $\\phi(N, \\delta, K)$ is said to be \\emph{admissible} if for arbitrary sets $A, B \\subset \\mathbb{Z}^{[n]}$ and a graph $G$ on $A \\times B$ satisfying (\\ref{eq:setupNGKdelta1})-(\\ref{eq:setupNGKdelta3}) the following holds.\n\nThere is a graph $G' \\subseteq G$ such that\n\n\\begin{enumerate}[(i)]\n\n\\item[(G)]{Graph size is controlled by $\\phi$:}\n$$\n\t|G'| \\geq \\phi(N, \\delta, K)\n$$\n\n\\item[(S)]{Separation of $G'$-neigborhoods is controlled by $\\psi$:\\\\} \nFor any $a \\in A$ (resp. $b \\in B$) the $\\mathcal{P}$-preimage of the $G'$-neighborhood \n$$\n\\mathcal{P}^{-1}\\left[G'(a)\\right] := \\mathcal{P}^{-1}\\left[\\{b \\in B: (a, b) \\in G' \\} \\right].\n$$ (resp. of $G'(b)$) is $\\psi(N, \\delta, K)$-separating.\n\\end{enumerate}\nFurthermore, we will assume that the following technical conditions hold for $\\phi(N, \\delta, K), \\psi(N, \\delta, K)$:\n\\begin{enumerate}\n \\item[(A1)] $\\phi, \\psi$ are non-decreasing in $N$\n \\item[(A2)] $\\phi$ is non-decreasing in $\\delta$, non-increasing in $K$ and for each $\\delta$ and $K$, we have $\\phi(N, \\delta, K) \\leq N$.\n \\item[(A3)] $\\psi$ is non-decreasing in $K$\n \\item[(A4)] If $N \\geq M$ then\n $$\n \\frac{\\phi(N, \\delta, K)}{N} \\leq \\frac{\\phi(M, \\delta, K)}{M}\n $$\n\\end{enumerate}\n\\end{Definition}\n\n\nNote that, by Claim \\ref{cor:trivialseparation}, \nthe pair $\\psi(N, \\delta, K) := N; \\phi(N, \\delta, K) := \\delta N$ is trivially admissible with much room to spare.\n\nThe following lemma gives a Freiman-type pair of admissible functions which is better than trivial in the regime $K=o(\\log N)$, and will be used later to bootstrap the argument. \n\n\\begin{Lemma}[Freiman-type admissible functions] \\label{lm:FreimanAdmissiblePair}\nThere is an absolute constant $C > 0$ such that the pair of functions\n\\begin{enumerate}\n\t\t\\item $\\psi(N, \\delta, K) := \\min \\left\\{ (2k^2)^{\\left(\\frac{K}{\\delta}\\right )^C}, N \\right\\}$ \\label{eq:Freimanadmissiblepsi} \n\t\t\\item $\\phi(N, \\delta, K) := \\left( \\frac {\\delta}{K} \\right)^C N$  \\label{eq:Freimanadmissiblephi}\n\\end{enumerate}\nis admissible.\n\\end{Lemma} \n\\begin{proof}\nThis pair is easily seen to satisfy (A1) through (A4). Thus it remains to check (G) and (S). By the setup, we are given two sets $\\mathcal{A}$ and $\\mathcal{B}$ of sizes $N_A$ and $N_B$ respectively, and a graph $G$ of size $\\delta N_AN_B$ such that\n\\begin{equation}\n |A +_G B| \\leq K \\sqrt{N_AN_B} \\label{eq:A_1A_2Gdoubling}\n\\end{equation}\n\nAssume without loss of generality that $N_A \\geq N_B$ and take $X = A \\cup B$, which is of size  $ \\approx N_A$. Since by (\\ref{eq:A_1A_2Gdoubling})\n$$\n\\frac{K^2}{\\delta^2} N_B \\geq N_A\n$$\nwe have \n$$\n|G| \\gg \\frac{\\delta^3}{K^2}|X|^2\n$$\nand\n$$\n|X +_G X| \\ll K |C|. \n$$\nBy a variant of the Balog-Szemer\\'edi-Gowers theorem (see e.g. \\cite{TV}, Exercise 6.4.10) there is $X' \\subseteq X$ such that $|X' + X'| < K'|X'|$ and $|G \\cap (X' \\times X')| > \\delta' N_A^2$ with\n\\begin{align}\n\t \\delta' &> \\left( \\frac{\\delta}{K}\\right)^C \\label{eq:deltaboundFreiman}\n\\\\\t K' &< \\left( \\frac{K}{\\delta} \\right)^C. \\label{eq:KboundFreiman}\n\\end{align}\n\nBy Theorem \\ref{thm:FreimanLemma} any subset of $X$ has rank at most $K'$ and by Theorem \\ref{thm:chang2}, the $\\mathcal{P}$-preimage of any subset of $X'$ is at most $(2k^2)^{K'^C}$-separating for some $C > 0$.  Thus, taking $G' := G \\cap (X' \\times X')$ by (\\ref{eq:deltaboundFreiman}) and (\\ref{eq:KboundFreiman}) we verify that the pair (\\ref{eq:Freimanadmissiblepsi}), (\\ref{eq:Freimanadmissiblephi}) is admissible.\n\\end{proof}\n\nThe goal is to find a better pair of admissible functions. The lemma below implements the `induction on scales' approach, which allows one to cook up a new pair $\\phi_*(N, \\cdot, \\cdot), \\psi_*(N, \\cdot, \\cdot)$ from a given pair of admissible functions, but taken at the smaller scale  $\\approx N^{1\/2}$.\n\n\n\n\n \n\\begin{Lemma} \\label{lm:InductionStepLemma}\n\nLet $\\psi$ and $\\phi$ be an admissible pair of functions. Then for some absolute constant $C > 0$ the pair of functions\n\\ben \\label{eq:admissiblepairinduction}\n\t\t\\psi_{*}(N, \\delta, K) &:=& Ck^2 \\max  \\psi (N', \\delta', K') \\psi  (N'', \\delta'', K'') \\\\\n\t\t\\phi_{*}(N, \\delta, K) &:=& \\min \\phi(N', \\delta', K') \\phi(N'', \\delta'', K'') \\label{eq:admissiblepairinduction2}\n\\een\nis admissible.\n\n Here $\\min$ and $\\max$ is taken over the data $(N', \\delta', K'), (N'', K'', \\delta'')$ such that\n\\ben  \\label{eq:parameterconstraints}\n\t\\lr{c\\frac{\\delta^9}{\\log^{22} (K\/\\delta)}} N &\\leq& N'N'' \\leq N\\\\\n    N' + N'' &\\leq& \\lr{C \\frac{K^{11}}{\\delta^{45}}}N^{1\/2}\\\\\n\tK'K'' &\\leq& \\lr{C \\frac{\\log^{15} K}{\\delta^{20}}}K\\\\\n\t\\delta' \\delta'' &\\geq& \\lr{c \\frac{1}{\\log^{6} (K\/\\delta)}} \\delta. \n\\een\n\n\\end{Lemma}\n\n\\begin{proof}\n\tLet us first check that $(\\phi_*, \\psi_*)$ given by (\\ref{eq:admissiblepairinduction}) and (\\ref{eq:admissiblepairinduction2}) indeed satisfy (A1) through (A4). Assume $N_1 < N_2$ and $\\delta, K$ are fixed. Then $\\psi_*(N_1, \\cdot, \\cdot) < \\psi_*(N_2, \\cdot, \\cdot)$ since for $\\psi_*(N_2, \\cdot, \\cdot)$ the maximum is taken over the larger range of parameters \n    $$\n    N'N'' \\leq N_2, \\,\\,\\,\\, N' + N'' \\leq C \\delta^{-45} K^{11}N_2^{1\/2}.\n    $$\n  Similarly, \n  $$\n  \\phi_*(N_1, \\cdot, \\cdot) < \\phi_*(N_2, \\cdot, \\cdot)\n  $$ \n  since the minimum is now taken over the smaller set \n  $$\n  c \\delta^9 \\log^{-22} (K\/\\delta) N_2 \\leq N'N''.\n  $$\n Note, that here we have used the fact that $\\phi$ and $\\psi$ are both increasing. This proves (A1).\n \n    In order to prove (A2) it suffices to note that when $\\delta$ increases (resp. $K$ decreases) the range of parameters $N', N'', \\delta', \\delta'', K', K''$ over which the minimum in $\\phi_*$ is taken is getting more narrow. Similarly, when $K$ increases the maximum in $\\psi_*$ is taken over a larger set which proves (A3). \n    \n\tIt remains to verify (A4). Let $M, \\delta, K$ be fixed and $M', M'', \\delta', \\delta'', K', K''$ be such that the minimum for $\\phi_*(M, \\delta, K)$ in (\\ref{eq:parameterconstraints}) is achieved. Let $c > 0$ be a parameter. Then $cM', cM'', \\delta', \\delta'', K', K''$ are in the admissible range for $\\phi_*(c^2M, \\delta, K)$ so\n    \\begin{align*}\n    \\phi_*(c^2M, \\delta, K) &\\leq \\phi(cM', \\delta', K')\\phi(cM'', \\delta'', K'') \\\\\n    &\\leq c^2\\phi(M', \\delta', K')\\phi(M'', \\delta'', K'') \\\\\n    &= c^2\\phi_*(M, \\delta, K).\n\t\\end{align*}\n Taking $c$ such that $c^2M = N$ we get (A4).\n \n\tLet $A, B\\subset \\mathbb{Z}^{n}$ of sizes $N_A, N_B$ respectively, $G \\subseteq A\\times B$ and suppose that the conditions (\\ref{eq:setupNGKdelta1})-(\\ref{eq:setupNGKdelta3}) are satisfied with parameters $(N, \\delta, K)$ where $N=N_AN_B$. Our ultimate goal is to find a subgraph of $G$ of size at least \n    \\[ \\phi(N', \\delta', K') \\phi(N'', \\delta'', K'') \\] \nsuch that the $\\mathcal{P}$-preimage of any its neighbourhoods is \n\\[Ck^2 \\psi (N', \\delta', K') \\psi  (N'', \\delta'', K'')-\\text{separating},\\] for some $N',N'',K',K'',\\delta ', \\delta ''$ satisfying \\eqref{eq:parameterconstraints}. Once this is done, the proof will be complete.  In order to achieve this goal, we will apply Lemma \\ref{lm:FiberingLemma} and then use the hypothesis that the pair $\\psi, \\phi$ is admissible for much smaller sets.\t\n \t\nDefine a function $f(t)$ for $0 \\leq t \\leq n$ as\n$$f(t)=\\max_{(a_1,b_1)\\in \\pi_{[t]}(A)\\times\\pi_{[t]}(B)}\\{|A_2(a_1)|+|B_2(b_1)|\\},$$ where $\\pi_{[t]}$ is the projection onto the first $t$ coordinates, and $A_2(a_1)$ and $B_2(b_1)$ are the fibres above $a_1$ and $b_1$ respectively.\nNote that $f$ is decreasing, $f(0)=|A|+|B|\\geq N^{1\/2}$, and $f(n) = 0$. Thus there is $t'$ such that\n \t\\ben \\label{eq:FirstcoordinateSplit}\n\t\tf(t') \\geq N^{1\/4} \t\n \t\\een\n\tbut\n\t\\ben \\label{eq:FirstcoordinateSplit2}\n\t\tf(t'+1) < N^{1\/4} \t.\n   \\een\t\n   \n\tWe use the $t'$ defined above for the decomposition $\\ZZ^{n}=\\ZZ^{t'}\\times \\ZZ^{n-t'}$ and let $\\pi_1$ and $\\pi_2$ denote the projection onto the first and second factor respectively. We now apply Lemma \\ref{lm:FiberingLemma} and get sets $A'\\subseteq A$ and $B'\\subseteq B$ together with a graph $G' \\subseteq G\\cap(A'\\times B')$ such that \n \t\\ben\n \t\tA' &=& \\bigcup_{a_1\\in \\pi_1(A') } a_1\\oplus A'_2(a_1) \\\\\n \t\tB' &=& \\bigcup_{b_1\\in \\pi_1(B') } b_1\\oplus B'_2(b_1)\n \t\\een\n and the fibers $A'_2(a_1), B'_2(b_1)$ together with the fiber graphs $G'_2(a_1,b_1)$ are uniform as defined in the statement of Lemma \\ref{lm:FiberingLemma}. Note that it is possible that $t' = 0$, in which case the sets split trivially with $\\pi_1(A') = \\pi_1(B') = \\{0\\}$. \n \n Using the notation of Lemma \\ref{lm:FiberingLemma}  we have \n \\ben\n \t|\\pi_1(A')+_{G'_1} \\pi_1(B)| \\leq K_1(M_AM_B)^{1\/2}.\n \\een\nSince $\\phi, \\psi$ is an admissible pair, there is $G''_1 \\subseteq G'_1$ of size at least $\\phi(M_1M_2, \\delta_1, K_1)$ such that all $\\mathcal{P}$-preimages of its vertex neighbourhoods are $\\psi(M_1M_2, \\delta_1, K_1)$-separating.\nNext, since $G_1''\\subseteq G_1'$, for each edge $(a_1, b_1) \\in G''_1$, there is a graph $G'_2(a_1, b_1)\\subseteq A'_2(a_1) \\times B'_2(b_1)$ such that $|G'_2(a_1, b_1)| \\geq \\delta_2m_Am_B$ and\n\\ben\n\t|A'_2(a_1) +_{G'_2(a_1, b_1)} B'_2(b_1)| \\leq K_2(m_Am_B)^{1\/2}.\n\\een\nAgain, by admissibility of $\\phi, \\psi$, there is $G''_2(a_1, b_1) \\subseteq G'_2(a_1, b_1)$ of size at least $\\phi(m_Am_B, \\delta_2, K_2)$ such that all $\\mathcal{P}$-preimages of its vertex neighbourhoods are $\\psi(m_Am_B, \\delta_2, K_2)$-separating.\n \t\nNow define $G'' \\subseteq G \\cap (A' \\times B')$ as \n$$\n\tG'' := \\{(a_1 \\oplus a_2, b_1 \\oplus b_2): (a_1, a_2) \\in G''_1, (a_2, b_2) \\in G''_2(a_1,b_1)\\}.\n$$\nIt is clear by construction that indeed all vertices of $G''$ belong to $A'$ and $B'$ respectively. Moreover, we have\n\\ben \\label{eq:firstreduction}\n\t|G''| \\geq \\phi(M_AM_B, \\delta_1, K_1)\\phi(m_Am_B, \\delta_2, K_2).\n\\een\n \t\nNow let's estimate the separating constant for the $\\mathcal{P}$-preimage of a neighbourhood $\\mathcal{P}^{-1}[G''(u)]$ of some $u \\in V(G'')$. Without loss of generality assume that $n \\in B'$ and $b = b_1 \\oplus b_2$. We can write\n\\ben\n\t\tG''(b) = \\bigcup_{a_1 \\in G''_1(b_1)} \\bigcup_{a_2 \\in G''_2(a_1, b_1)}\\{a_1 \\oplus a_2\\} .\n\\een\nThus,\n\\ben \n\\mathcal{P}^{-1}[G''(b)] = \\bigcup_{a_1\\in G''_1(b_1)} p_1^{a_1} \\cdot \\left\\{ \\bigcup_{a_2 \\in G''_2(a_1, b_1)} p_2^{a_2} \\right\\}.\n\\een\nHere we are using the notation $q^{r}=q_1^{r_1}\\cdots q_l^{r_l}$ for a vector $q$ of primes and a vector $r$ of integers, and $p_1$ and $p_2$ are respectively the first $t$ primes from the map $\\cP$ and the remaining primes. Now, since $G''_1(b_1)$ and $G''_2(a_1, b_1)$ are orthogonal as linear sets we conclude that $(p_1^{a_1}, p_2^{a_2}) = 1$.  Thus, by Lemma \\ref{lem:chain} and the admissibility of $\\phi, \\psi$ applied to $G''_1$ and $G''_2(a_1,b_1)$ we conclude that $\\mathcal{P}^{-1}[G''(b)] $ is at most $\\psi(M_AM_B, \\delta_1, K_1)\\psi(m_Am_B, \\delta_2, K_2)$-separating.\n\nWe now record the bounds for the various parameters following from Lemma \\ref{lm:FiberingLemma}. We have \n\\ben\n\t\\delta_1\\delta_2 &\\geq& \\lr{c \\frac{1}{\\log^{3}(K\/\\delta)}}\\delta. \\label{eq:delta_2}\\\\\n\tK_1 K_2 &\\leq&  \\lr{C\\frac{\\log K }{\\delta^{2}}}K   \\label{eq:K_2} \\\\\n\tM_Am_A &\\geq & \\lr{c \\frac{\\delta^2}{\\log(K\/\\delta)}}N_A \\label{eq:M_1m_1} \\\\\n\tM_Bm_B &\\geq & \\lr{c \\frac{\\delta^2}{\\log(K\/\\delta)}} N_B \\label{eq:M_2m_2} \\\\\n\tm_A, m_B &\\geq & \\lr{c \\frac{\\delta^{10}}{K^{4}}} N^{1\/4} \\label{eq:m_ibig}\n\\een\n\nIn particular, we have\n\\ben \\label{eq:M_1M_2upperbound}\n M_AM_B < \\frac {N_AN_B}{m_Am_B} < \\lr{c \\frac{K^8}{\\delta^{20}}}N^{1\/2}.\n\\een\nAs a first attempt, we set $N'=M_AM_B$ and $N''=m_Am_B$, $\\delta'=\\delta_1$, $K'=K_1$, $\\delta''=\\delta_2$ and $K''=K_2$.\nIf $N''=m_Am_B$ is less than $N^{1\/2}$, one can verify that all of the above bounds comply with the statement of this lemma, and we can stop. If $N''$ is too big, we will apply Lemma \\ref{lm:FiberingLemma} again.\n\nTo further reduce the size we apply Lemma \\ref{lm:FiberingLemma} again for each pair of sets $(A'_2(a_1), B'_2(b_1))$ such that $(a_1, b_1) \\in G'_1$, stripping off only a single coordinate as explained below. Assume the base point $(a_1, b_1)$ is fixed henceforth. \n\n\\begin{comment}\nThe reader may keep in mind the following model case: $N_1 = N_2 = N^{1\/2}$ and $A_1, A_2$ are three-dimensional arithmetic progressions\n$$\n\\{ [0,N_1^\\alpha]e_1 + [0, N_1^{1-2\\alpha}]e_2 +  [0, N_1^\\alpha]e_3  \\},\n$$\nfor some small $\\alpha > 0$. In this case $t'=1, M_1 M_2 \\approx N^\\alpha$  and $m_1m_2 \\approx N^{1-\\alpha}$. However, the fibers $A_1(x)$ and $A_2(y)$ are heavily concentrated on the second coordinate which makes the separating constant of the $\\mathcal{P}$-preimage small.\n\\end{comment}\n\\vskip 1em\n\nWe split the coordinates $\\{t'+1, \\ldots, n \\}$ as  $\\ZZ\\times \\ZZ^{n-t'-2}$. We apply Lemma \\ref{lm:FiberingLemma}, this time with to the pair of sets $A'_2(a_1)$ and $B'_2(b_1)$ and the graph $G'_2(a_1,b_1)$. To ease notation, let us set $U=A'_2(a_1)$, $V=B'_2(b_1)$, and $H=G'_2(a_1,b_1)$. Here, it is worth noting that $U,V$ and $H$ depend on the base point $(a_1,b_1)$. This time, we have the estimates\n\\[|U|\\approx m_A, |V|\\approx m_B\\] and\n\\[|U+_HV|\\leq K_2\\lr{|U||V|}^{1\/2}\\]\nwhere $|H|\\geq \\frac{\\delta_2}{4}|U||V|$.\nWe will again denote by $\\pi_1$ the projection onto the first coordinate, and by $\\pi_2$ the projection onto the remaining $n-t'-2$ coordinates. \nWe then get \n$$\nU' \\subseteq  U, \\,\\,\\, V'\\subseteq V\n$$ such that\n\\ben\n\tU' &=& \\bigcup_{u_1 \\in \\pi_{1}(U')} u_1\\oplus U'_2(u_1) \\\\\n\tV' &=& \\bigcup_{y_1 \\in \\pi_{1}(V')} v_1\\oplus V'_2(v_1)\n\\een\nand the fibers $U'_2(u_1)$ and $V'_2(v_1)$ are of approximately the same size, say $m_U$ and $m_V$ respectively. We also write $M_U=|\\pi_1(U)|$ and $M_V=|\\pi_1(V)|$. Note again that, for instance,  the fiber $U'_2(u_1)$ may be trivial (i.e. $\\{ 0\\}$), which simply means that $m_U \\approx 1$. By (\\ref{eq:FiberingLemmaSetSizes}), (\\ref{eq:FiberingLemmaSetSizes2}) we have the estimates\n\\[M_Um_U\\geq c\\delta_2^2(\\log(K_2\/\\delta_2))^{-1}|U|,\\ M_Vm_V\\geq c\\delta_2^2(\\log(K_2\/\\delta_2))^{-1}|V|\\]\n\nNext, we have a graph \\[H'\\subseteq (U'\\times V')\\cap H\\] with uniform fibers as defined in Lemma \\ref{lm:FiberingLemma}. The graph $H'$ splits into the base graph $H'_{1} \\subset \\pi_{1}(U') \\times \\pi_{1}(V')$ such that\t\n\\[|\\pi_1(U')+_{H'_1}\\pi_1(V')|\\leq K_3(M_UM_V)^{1\/2},\\]\nand fiber graphs $H'_2(u_1, v_1)$ such that for $(u_1, v_1) \\in H_{1}'$ \n\\ben\n|U'_2(u_1) +_{H'_2(u_1,v_1)} V'_2(v_1)| \\leq K_4(m_Um_V)^{1\/2},\n\\een\nwith\n\\ben\n|U'_2(u_1)| &\\approx& m_U \\\\\n|V'_2(v_1)| &\\approx& m_V \\\\\n|H'_2(u_1, v_1)| &\\geq& \\delta_4 m_Um_V.\n\\een\n\nThe parameters $m_U, m_V, \\delta_3, \\delta_4, K_3, K_4$ as well as the sizes of $H_1'$ and $H'_2(u_1,v_1)$ are controlled by Lemma \\ref{lm:FiberingLemma}. By the assumption that the original pair $(\\phi,\\psi)$ is admissible, for each such a graph $H'_2(u_1,v_1)$ there is a subgraph $H_2''(u_1,v_1) \\subseteq H_2'(u_1,v_1)$ with\n\\ben\n|H_2''(u_1,v_1)| \\geq \\phi(m_Um_V, \\delta_4, K_4)\n\\een\nsuch that the $\\mathcal{P}$-preimage of each neighborhood of $H_2''(u_1,v_1)$ is $\\psi(m_Um_V, \\delta_4, K_3)$-separating. Define $H'' \\subset H'$ as\n\\ben\n\tH''= \\{(u_1 \\oplus u_1, v_1 \\oplus v_2): (u_1, v_1) \\in H'_1, (u_2,v_2)\\in H_2''(u_1,v_1) \\}. \n\\een\nThe size of $H''$ is at least $|H_1'|\\phi(m_Um_V, \\delta_4, K_4)$. Next, the set of vertices of $H_1'$ all lie in a one-dimensional affine subspace, so combining Corollary \\ref{cor:oneprimeseparation} and Lemma \\ref{lem:chain} one concludes that the $\\mathcal{P}$-preimage of each neighborhood of $H''$ is \n$Ck^2\\psi(m_Um_V, \\delta_4, K_4)$-separating with some absolute constant $C > 0$. Putting together all of the details, we conclude\nthat, for $G_2'(a_1,b_1)\\subset A'(a_1) \\times B'(b_1)$, there is a subgraph $H'' \\subseteq G_2'(a_1,b_1)$ of size at least\n\\ben \\label{eq:step2psiphi}\n\t\\phi_{a_1, b_1} := |H_1'|\\phi(m_Um_V, \\delta_4, K_4)  \n\\een\nsuch that the $\\mathcal P$-preimage of each neighbourhood in $H''$ is $\\psi_{a_1,b_1}$-separating, where\n\\[\\psi_{a_1, b_1} := Ck^2\\psi(m_Um_V, \\delta_4, K_4).\\]\nSince the the graph $H''$ depends on the pair $(a_1,b_1)$, we now rename this graph $H''_{a_1,b_1}$.\n\nIn turn, substituting $\\psi_{a_1, b_1}$ and $\\phi_{a_1, b_1}$ into the argument leading to (\\ref{eq:firstreduction}) and Lemma \\ref{lem:chain}, we construct a graph\n\\[G''':= \\{(a_1 \\oplus a_2, b_1 \\oplus b_2) :(a_1,b_1) \\in G_1', (a_2,b_2) \\in H''_{a_1,b_1}\\}.\\] \nThe graph $G'''$ has size at least\n\\begin{equation}\n\\phi(M_AM_B, \\delta_1, K_1) \\cdot \\min_{(a_1, b_1) \\in G'_1} \\phi_{a_1, b_1},\n\\label{eq:admissiblepairs2}\n\\end{equation}\nand the separating factors are at most\n\\begin{equation}\n\\psi(M_AM_B, \\delta_1, K_1) \\cdot   \\max_{(a_1, b_1) \\in G'_1} \\psi_{a_1, b_1},\n\\label{eq:admissiblepairs22}\n\\end{equation}\n \nWith $G'''$ we have now found a large subgraph with good separating factors. In the remaining calculations, we show that the existence of this $G'''$ is good enough to imply the theorem. Essentially it remains to check that the quantities (\\ref{eq:admissiblepairs2}) and \\eqref{eq:admissiblepairs22} can indeed be bounded respectively by \\eqref{eq:admissiblepairinduction2} and (\\ref{eq:admissiblepairinduction}).\nNote that the quantities (\\ref{eq:admissiblepairs2}) and \\eqref{eq:admissiblepairs22} do depend on the structure of $A$ and $B$. We are going to show, however, that they are uniformly bounded by \\eqref{eq:admissiblepairinduction2} and (\\ref{eq:admissiblepairinduction}) which are functions of $(N, \\delta, K)$ only. We remark here that we will make use of the following fact: if $|X+_GY|\\leq K(|X||Y|)^{1\/2}$ for some $G\\subset X\\times Y$ of size at least $\\delta|X||Y|$, then $K\/\\delta\\geq 1$. \n\n\nFirst, since $(a_1, b_1) \\in G'_1$ we have by (\\ref{eq:FiberingLemmaDeltaSizes})\n\\begin{equation}\n\t \\delta_4\\geq\\delta_3\\delta_4 >c\\log^{-3}(K_2\/\\delta_2) \\delta_2.\n     \\label{eq:delta3}\n\\end{equation}\nBy (\\ref{eq:FiberingLemmaDoublingConstants}) and (\\ref{eq:FiberingLemmaDeltaSizes})\n\\ben\n\t\\frac {K_2} {\\delta_2}\\leq \\frac{K_1K_2}{\\delta_1\\delta_2} < \\frac{CK \\log(K) \\log^3(K\/\\delta)} {\\delta^{3}}\n\\een\n and so\n\\begin{equation}\n \\log (K_2\/\\delta_2) < C\\log(K\/\\delta).\n \\label{eq:Koverdelta}\n\\end{equation}\nConsequently,\n\\ben \\label{eq: delta_1delta_3}\n \\delta_1 \\delta_4 \\stackrel{\\eqref{eq:delta3}, \\eqref{eq:Koverdelta}}{>} c \\log^{-3} (K\/\\delta) \\delta_1 \\delta_2 \\stackrel{\\eqref{eq:delta_2}}{>} c \\log^{-6} (K\/\\delta) \\delta.\n\\een\nNext, by (\\ref{eq:FiberingLemmaDoublingConstants})\n\n\\begin{equation}\n K_4  \\leq \\frac{K_3K_4}{\\delta_3\\delta_4}\\leq C K_2 \\log^2 (K_2) \\delta_2^{-4}\n \\label{eq:K3K2}\n\\end{equation}\nand by (\\ref{eq:FiberingLemmaDeltaSizes})\n\\ben\n\\delta_2 &>& c \\log^{-3}(K\/\\delta) \\delta \\label{eq:delta2delta}\\\\\nK_2 &<& C  \\delta^{-4} K \\log^2 K .\n\\een\nTherefore\n\\ben\n\\log^2(K_2) \\delta_2^{-4} &\\leq& C \\log^{14} (K\/\\delta) \\delta^{-4} \\label{eq:logK2delta2}  \\\\\n\t\t\t\t\t\t\t\t\t&=& C (\\delta^{14} \\log^{14} (K\/\\delta)) \\delta^{-18} < C (\\log^{14}K) \\delta^{-18} \\nonumber\n\\een\nand\n\\ben \\label{eq:K_1K_3bound}\nK_1 K_4 \\stackrel{\\eqref{eq:K3K2}}{\\leq} C K_1 K_2 \\log^2 (K_2) \\delta_2^{-4} \\stackrel{(\\ref{eq:K_2}), (\\ref{eq:logK2delta2})}{\\leq} C \\frac{K \\log^{15} K}{\\delta^{20}}.\n\\een\nFinally, we have by (\\ref{eq:FiberingLemmaSetSizes}), (\\ref{eq:FiberingLemmaDeltaSizes}), (\\ref{eq:FiberingLemmaBaseGraphSize}) and (\\ref{eq:FiberingLemmaFiberGraphSize}) that\n\\ben\n|H_1'|m_Um_V &\\geq& c \\log^{-3}(K_2\/\\delta_2) \\delta_2 (\\delta_2^4 \\log^{-2}(K_2\/\\delta_2)) |A_2(a_1)||B_2(b_1)| \\nonumber \\\\\n\t\t\t\t\t&\\geq& c\\log^{-5} (K\/\\delta) \\delta_2^5 m_Am_B \\stackrel {(\\ref{eq:delta2delta})}{\\geq} c \\log^{-20} (K\/\\delta) \\delta^5 m_Am_B  \\label{eq:K_Il_1l_2} .\n\\een\n\nDefine \n\\ben \\label{eq:N''def}\n N'' := \\min \\{ N^{1\/2},  \\max\\{m_Um_V, c \\log^{-20} (K\/\\delta) \\delta^5 m_Am_B\\} \\}.\n\\een\nBy our choice of $t'$ it follows that $m_Um_V \\leq N''$. By (A4) we have\n\\ben\n\\frac{m_Um_V}{N''}\\phi(N'', \\delta_3, K_3) \\leq  \\phi(m_Um_V, \\delta_4, K_4).\n\\een\nDefining \n\\ben\nN' := \\frac{M_AM_Bm_Um_V}{N''}|H_1'|,\n\\een\nwe have by (\\ref{eq:K_Il_1l_2}) and (\\ref{eq:N''def}) that $M_AM_B \\leq N'$, so by (A4) again\n\\ben\n\\frac{M_AM_B}{N'}\\phi(N', \\delta_1, K_1) \\leq  \\phi(M_AM_B, \\delta_1, K_1),\n\\een \nso\n\\ben \\label{eq:phiNprimesbound}\n\t\\phi(N', \\delta_1, K_1) \\phi (N'', \\delta_3, K_3) &\\leq& \\frac{N'}{M_AM_B}\\phi(M_AM_B, \\delta_1, K_1) \\frac{N''}{m_Um_V}\\phi(m_Um_V, \\delta_3, K_3) \\nonumber \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t &\\stackrel {(\\ref{eq:step2psiphi})}{=}&  \\phi(M_AM_B, \\delta_1, K_1) \\phi_{a_1,b_1}.\n\\een\nOn the other hand, \n\\ben \\label{eq:Nprimesproduct}\nN'N'' &=& M_AM_Bm_Um_V |H_1'| \\\\\n&\\stackrel {(\\ref{eq:K_Il_1l_2})}{\\geq}& c\\log^{-20} (K\/\\delta) \\delta^5 M_AM_Bm_Am_B \\\\\n\t\t&\\stackrel {(\\ref{eq:FiberingLemmaSetSizes}), (\\ref{eq:FiberingLemmaSetSizes2})}{\\geq}& c  \\delta^9 \\log^{-22} (K\/\\delta) N.\n\\een\nAlso, since\n\\ben\n\tm_Am_B \\stackrel{\\eqref{eq:m_ibig}}{>} c \\delta^{20} K^{-8} N^{1\/2},   \\nonumber\n\\een\nit follows from the definition of $N''$ in \\eqref{eq:N''def} that\n\\ben\n\tc \\delta^{45} K^{-11}N^{1\/2} \\leq N'' \\leq N^{1\/2} . \\nonumber\n\\een\nThen, since $N'' N' \\leq N$,\n\\ben\n  N' \\leq C \\delta^{-45} K^{11}N^{1\/2}, \\nonumber\n\\een\nand so\n\\ben \\label{eq:Nprimessum}\n N' + N'' \\leq C \\delta^{-45} K^{11}N^{1\/2}.\n\\een\n\nWe now have all the estimates to finish the proof. The bounds (\\ref{eq: delta_1delta_3}), (\\ref{eq:K_1K_3bound}), (\\ref{eq:Nprimesproduct}), (\\ref{eq:Nprimessum}) verify that the parameters \n\\ben \n\\delta' &:=& \\delta_1, \\,\\,\\, \\delta'' := \\delta_4 \\nonumber \\\\\nK' &:=& K_1, \\,\\,\\, K'' := K_4 \\nonumber \\\\\n\\een\nand $N', N''$ indeed satisfy the constraints (\\ref{eq:parameterconstraints}).  Recall that by (A1)  $\\psi(\\cdot, \\delta, K)$ is increasing in the first argument, so by (\\ref{eq:M_1M_2upperbound}) and (\\ref{eq:FirstcoordinateSplit2})\n\\ben \n\t\t\\psi_{*}(N, \\delta, K) \t&\\geq& Ck^2 \\psi\\lr{\\max\\left\\{N^{1\/2}, \\frac{N}{m_Am_B} \\right\\}, \\delta_1, K_1}  \\psi\\lr{\\min \\{ N^{1\/2}, m_Am_B \\}, \\delta_4, K_4} \\nonumber \\\\\n\t\t\t\t\t&\\geq&  \\psi(M_AM_B, \\delta_1, K_1) \\psi_{x, y} . \\label{eq:psistargeneralbound}\n\\een\nIn the previous inequality, we have used monotonicity (A1) and the information that $\\frac{N}{m_Am_B} \\geq M_AM_B $, $N^{1\/2} \\geq m_Um_V$, $m_A \\geq m_U$ and $m_B \\geq m_V$.\n\n\nAlso, (\\ref{eq:phiNprimesbound}) and (\\ref{eq:admissiblepairs2}) verify that \n\\ben \n\\phi_{*}(N, \\delta, K) &\\leq& \\phi(N', \\delta_1, K_1) \\phi (N'', \\delta_4, K_4)  \\nonumber \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t &\\leq& \\phi(M_AM_B, \\delta_1, K_1) \\phi_{a_1, b_1}. \\label{eq:phistargeneralbound}\n\\een\n It follows that the pair $(\\psi_{*}, \\phi_{*})$ is indeed admissible since (\\ref{eq:psistargeneralbound}) and (\\ref{eq:phistargeneralbound}) hold for all base points $(a_1, b_1) \\in G'_1$ and thus uniformly bound \\eqref{eq:admissiblepairs22} and (\\ref{eq:admissiblepairs2}) respectively. \n\\end{proof}\n\n\\section{A better admissible pair} \\label{sec:betterpair}\n\nWith Lemma \\ref{lm:InductionStepLemma} at our disposal we can start with the data $(N, \\delta, K)$ and  reduce the problem to the case of smaller and smaller $N$ and $K$ with reasonable losses in $\\delta$. The process can be described by a binary a tree where each node with the data $(N, \\delta, K)$ splits into two children with the attached data being approximately equal to $(N^{1\/2}, \\delta', K')$ and $(N^{1\/2}, \\delta', K'')$, with $K'K''$ roughly equal to $K$ and $\\delta'\\delta''$ roughly equal to $\\delta$. Thus, when the height of the tree is about $\\log \\log K$, the $K$'s in the most of the nodes should be small enough so that Lemma \\ref{lm:FreimanAdmissiblePair} becomes non-trivial. Going from the bottom to the top we then recover an improved admissible pair of functions at the root node. \n\n\\begin{Lemma} \\label{lm:beteradmissiblepair}\nFor any $\\gamma > 0$ there exists $C(\\gamma) > 0$ such that the pair\n\\ben\n\t\\phi(N, \\delta, K) := \\left( \\frac{\\delta}{K}\\right)^{C \\log \\log (K\/\\delta)} N \\label{eq:betterphi}\\\\\n\t\\psi(N, \\delta, K) := k^{ \\log (K\/\\delta)^{C\/\\gamma}} N^{\\gamma}  \\label{eq:betterpsi}\n\\een  \nis admissible. \n\\end{Lemma}\n\n\\begin{proof}\nLet $N, \\delta, K$ be fixed. Take an integer $t = 2^l$ to be specified later ($l$ is going to be the height of the tree and $t$ the total number of nodes). \n\nLet $(\\phi_0, \\psi_0)$ be the Freiman-type admissible pair given by Lemma~\\ref{lm:FreimanAdmissiblePair}. We apply recursively Lemma~\\ref{lm:InductionStepLemma} and obtain admissible pairs for $i = 1, \\ldots, l$ as follows\n\n\\ben \\label{eq:admissiblepairinductionleveli}\n\t\t\\psi_{i} &:=& \\max Ck^2 \\psi_{i-1} (N', \\delta', K') \\psi_{i-1}  (N'', \\delta'', K'') \\\\\n\t\t\\phi_{i} &:=& \\min \\phi_{i-1}(N', \\delta', K') \\phi_{i-1}(N'', \\delta'', K''),\n\\een\n(with the $\\max$ and $\\min$ taken over the set of parameters constrained by (\\ref{eq:parameterconstraints})). Thus, at the root node we have the admissible pair $\\psi := \\psi_{l-1}, \\phi := \\phi_{l-1}$ given by\n\\ben\n\t\\psi(N, \\delta, K) &:=& (Ck^2)^{2^l} \\prod_{\\nu \\in \\{0, 1 \\}^l} \\psi_0(N'_\\nu, \\delta'_\\nu, K'_\\nu)\t\\label{eq:betterpsiproductformula}\\\\\n\t\\phi(N, \\delta, K) &:=& \\prod_{\\nu \\in \\{0, 1 \\}^l} \\phi_0(N_\\nu, \\delta_\\nu, K_\\nu) \\label{eq:betterphiproductformula} \n\\een\nfor some data $(N_\\nu, \\delta_\\nu, K_\\nu)$ and (possibly different) $(N'_{\\nu}, \\delta'_{\\nu}, K'_{\\nu})$ at the leaf nodes of the tree which attain the respective maxima and minima. For intermediate tree nodes $\\nu$, denoting by $\\{\\nu, 0\\}$ and $\\{\\nu, 1\\}$ the left and right child of $\\nu$ respectively, one has\n\\begin{align}  \\label{eq:parameterconstraintslevel}\n   c_1 \\delta_\\nu^9 \\log^{-22} (K_\\nu\/\\delta_\\nu) N_\\nu &\\leq N_{\\nu, 0}N_{\\nu, 1} \\leq N_\\nu \\\\ \n   N_{\\nu, 0} + N_{\\nu, 1} &\\leq C_1 \\delta_\\nu^{-45} K_\\nu^{11}N_\\nu^{1\/2}\t \\label{eq:Nsquarerootsplitting} \\\\\n   K_{\\nu, 0}K_{\\nu, 1} &\\leq C_1 \\frac{\\log^{15} K_\\nu }{\\delta_\\nu^{20}} K_\\nu \\label{eq:Ksquarerootsplitting}  \\\\ \n   \\delta_{\\nu, 0} \\delta_{\\nu, 1} &\\geq c_1 \\log^{-6} (K_\\nu\/\\delta_\\nu) \\delta_\\nu, \\label{eq:deltasquarerootsplitting}\n\\end{align}\nand similarly for $(N'_{\\nu}, \\delta'_{\\nu}, K'_{\\nu})$. The absolute constants $c_1$ and $C_1$ are exactly those given in the statement of Lemma \\ref{lm:InductionStepLemma} as $c$ and $C$ respectively. They have been relabelled here in an attempt to distinguish them.\n\nIn what follows we assume that $N$ is large enough so that $\\log K_\\nu > C$ and $\\log(\\delta_\\nu^{-1}) > c^{-1}$ and the constants $C, c$ can be swallowed by an extra power of $\\log (K\/\\delta)$.\n\nWe have\n\\[\n \\log \\frac{K_{\\nu, 0}}{\\delta_{\\nu, 0}} +  \\log \\frac{K_{\\nu, 1}}{\\delta_{\\nu, 1}} < 20 \\log \\frac{K_\\nu}{\\delta_\\nu}\n\\]\nso for an arbitrary $1 < l' \\leq l$\n\\beq \\label{eq:Kdeltanubound}\n   \\max_{  \\nu \\in \\{0, 1\\}^{l'} } \\log \\frac{K_\\nu}{\\delta_\\nu}\n   \\leq \\sum_{  \\nu \\in \\{0, 1\\}^{l'} } \\log \\frac{K_\\nu}{\\delta_\\nu}\n   <  20^{l'} \\log \\frac{K}{\\delta}.\n\\eeq\nNext, it follows from \\eqref{eq:deltasquarerootsplitting} and (\\ref{eq:Kdeltanubound}) that \n\\begin{align} \n \\prod_{\\nu \\in \\{0, 1\\}^{l'}} \\delta_\\nu = \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} \\delta_{\\nu,0} \\delta_{\\nu,1} & \\geq  \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} c_1 \\left(\\log \\frac{K_\\nu}{\\delta_\\nu} \\right)^{-6} \\delta_\\nu \\nonumber\n \\\\&> \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} c_1 \\left(20^{l'} \\log \\frac{K}{\\delta}\\right)^{-6} \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} \\delta_\\nu \\nonumber\n \\\\& = \\left(\\frac{20}{c_1}\\right)^{-3l'\\cdot 2^{l'}} \\left(\\log \\frac{K}{\\delta}\\right)^{-3 \\cdot 2^{l'}} \\prod_{\\nu \\in \\{0, 1\\}^{l'-1}} \\delta_\\nu . \\label{eq:deltatoiterate}\n\\end{align}\nApplying \\eqref{eq:deltatoiterate} iteratively then yields\n\\begin{equation}\n \\prod_{\\nu \\in \\{0, 1\\}^{l'}} \\delta_\\nu > \\left (\\frac{20}{c_1} \\right)^{-6l' \\cdot 2^{l'}} \\left (\\log \\frac{K}{\\delta}\\right)^{-6\\cdot 2^{l'}} \\delta .\n\\label{eq:deltanuproductbound}\n\\end{equation}\nUsing similar arguments, we obtain the following bounds:\n\\beq \\label{eq:Knuproductbound}\n  \\prod_{\\nu \\in \\{0, 1\\}^{l'}} K_\\nu < \\left(\\frac{20C_1}{c_1}\\right)^{280 \\cdot l' 2^{l'}} \\left( \\log \\frac{K}{\\delta}\\right )^{280 \\cdot 2^{l'}} \\delta^{-20l'}K\n\\eeq\nand\n\\beq \\label{eq:Nnuproductbound}\n\t\\prod_{\\nu \\in \\{0, 1\\}^{l'}} N_\\nu > \\left(\\frac{20}{c_1}\\right)^{-160 \\cdot l' 2^{l'}} \\left(\\log \\frac{K}{\\delta} \\right)^{-160  \\cdot 2^{l'}} \\delta^{9 l'} N.\n\\eeq\nFor more details on how these bounds are obtained, see \\cite[p. 492]{BC}.\n\nSubstituting (\\ref{eq:deltanuproductbound}), (\\ref{eq:Knuproductbound}), (\\ref{eq:Nnuproductbound}) into (\\ref{eq:betterphiproductformula}) and Lemma \\ref{lm:FreimanAdmissiblePair} (\\ref{eq:Freimanadmissiblephi}) we get\n\\begin{align*} \n \\phi (N, \\delta, K) = \\prod_{\\nu \\in \\{0, 1\\}^{l}} \\phi_0(N_\\nu, \\delta_\\nu, K_\\nu)  \n &= \\prod_{\\nu \\in \\{0, 1\\}^{l}} \\left (\\frac{\\delta_\\nu}{K_\\nu}\\right )^C N_\\nu \n \\\\& \\geq  e^{-C'l2^l} \\left( \\log \\frac{K}{\\delta} \\right)^{-C'2^l} \\delta^{lC'} K^{-C'}N,\n\\end{align*}\nfor some suitable $C' > 0$. Taking\\footnote{Strictly speaking we should ensure that $l$ is an integer by taking $l := \\lfloor \\log \\log (K\/\\delta) \\rfloor$. In order to simplify calculations and avoid adding further multiplicative constants, we assume that $l$ as defined here is already an integer.}\n$$\nl := \\log \\log (K\/\\delta) \n$$ \nwe obtain\n\\[ \\phi (N, \\delta, K) \\geq \\left( \\frac{\\delta}{K}\\right)^{C \\log \\log (K\/\\delta)} N\\]\nfor some suitable $C>0$.\n\n\nWe now turn to $\\psi$. For the sake of notation we use again $(N_\\nu, \\delta_\\nu, K_\\nu)$ instead of $(N'_\\nu, \\delta'_\\nu, K'_\\nu)$. The bounds above, however, still hold.\n\nBy \\eqref{eq:betterpsiproductformula} and Lemma \\ref{lm:FreimanAdmissiblePair}\n\\begin{equation}\n\\psi(N,\\delta,K) =(Ck^2)^{2^l}\\prod_{\\nu \\in \\{0,1 \\}^l} \\min \\left\\{ (2k^2)^{\\left(\\frac{K_\\nu}{\\delta\\nu}\\right )^C}, N_\\nu \\right\\}. \n\\label{eq:psibound}\n\\end{equation}\n\n\nIn order to bound the quantity of the right hand side effectively, we will need a suitable uniform bound for individual $N_\\nu$, which we deduce below.\n\nIt follows from \\eqref{eq:deltasquarerootsplitting} that \n\\begin{equation}\n\\delta_{\\nu,0}, \\delta_{\\nu,1} \\geq c_1\\left(\\log \\frac{K_\\nu}{\\delta_\\nu}\\right)^{-6}\\delta_\\nu.\n\\label{eq:deltatriv}\n\\end{equation}\nApplying this bound as well as \\eqref{eq:Kdeltanubound}, it follows that for any $1 \\leq l' \\leq l$ and $\\nu \\in \\{0,1\\}^{l'}$,\n\\begin{equation} \\label{eq:anotherdeltabound}\n\\delta_\\nu = \\delta_{\\nu',\\cdot} \\geq c_1 \\left(\\log \\frac{K_{\\nu'}}{\\delta_{\\nu '}}\\right)^{-6}\\delta_{\\nu '} \\geq  (20C)^{-6l'} \\left(\\log \\frac{K}{\\delta}\\right)^{-6}\\delta_{\\nu '}.\n\\end{equation}\nIteratively applying \\eqref{eq:anotherdeltabound} yields\n\\begin{equation} \\label{eq:yetanotherdeltabound}\n\\delta_\\nu \\geq (20C)^{-6l'^2} \\left(\\log \\frac{K}{\\delta}\\right)^{-6l'}\\delta.\n\\end{equation}\n\nSimilarly, since $K_\\nu \\geq \\delta_\\nu$, it follows from \\eqref{eq:Ksquarerootsplitting} and \\eqref{eq:deltatriv} that\n\\[ K_{\\nu,0} \\leq C_1 \\frac{K_\\nu\\log^{15} K_\\nu}{\\delta_\\nu^{20}\\delta_{\\nu,1}}\n\\leq C_1' \\frac{\\left(\\log \\frac{K_\\nu}{\\delta_\\nu} \\right)^{21}}{\\delta_\\nu^{21}}K_\\nu.\n\\]\nThe same argument implies that $ K_{\\nu,1} \\leq C_1' \\frac{K_\\nu \\left(\\log \\frac{K_\\nu}{\\delta_\\nu} \\right)^{21}}{\\delta_\\nu^{21}}$. Therefore, by applying \\eqref{eq:Kdeltanubound} and \\eqref{eq:yetanotherdeltabound}, it follows that for any $\\nu \\in \\{0,1\\}^{l'}$,\n\\begin{equation} \\label{eq:anotherKbound}\nK_\\nu = K_{\\nu',*} \\leq C_1' \\frac{K_{\\nu'} \\left(\\log \\frac{K_{\\nu'}}{\\delta_{\\nu'}} \\right)^{21}}{\\delta_{\\nu'}^{21}} \\leq \\frac{(20C)^{147l'^2} \\left( \\log \\frac{K}{\\delta}\\right)^{147l'}}{\\delta^{21}} K_{\\nu'} .\n\\end{equation}\nIterating \\eqref{eq:anotherKbound} yields\n\\begin{equation}\nK_\\nu  \\leq \\frac{(20C)^{147l'^3} \\left( \\log \\frac{K}{\\delta}\\right)^{147l'^2}}{\\delta^{21l'}} K.\n\\label{eq:yetanotherKbound}\n\\end{equation}\n\nTo bound $N_\\nu$, first note that \\eqref{eq:Nsquarerootsplitting}, \\eqref{eq:yetanotherdeltabound} and \\eqref{eq:yetanotherKbound} together imply that for any $\\nu' \\in \\{0,1\\}^{l'} $,\n\\[ N_{\\nu, 0} + N_{\\nu, 1} \\leq C_1 \\delta_\\nu^{-45} K_\\nu^{11}N_\\nu^{1\/2}\t\\leq \\frac{(20C_1)^{1887 l'^3} \\left( \\log \\frac{K}{\\delta} \\right)^{1887l'^2}K^{11}}{\\delta^{276l'}}N_\\nu^{1\/2} .\\]\n Applying this bound iteratively yields (with some rather crude estimates)\n \\begin{equation}\n N_\\nu \\leq \\frac{(20C_1)^{4000 l'^3} \\left( \\log \\frac{K}{\\delta} \\right)^{4000 l'^2}K^{22}}{\\delta^{4000 l'}}N^{\\frac{1}{2^{l'}}}.\n \\label{eq:yetanotherNbound}\n \\end{equation}\n\nBefore inserting \\eqref{eq:yetanotherNbound} into \\eqref{eq:psibound}, we split the data $(N_\\nu, \\delta_\\nu, K_\\nu)$ into two parts, $I \\cup J = \\{0, 1 \\}^l$, such that\n$$\n\tI= \\left\\{  \\nu : \\frac{K_\\nu}{\\delta_\\nu} < T \\right \\}\n$$\nand\n$$\n\tJ= \\left \\{ \\nu:  \\frac{K_\\nu}{\\delta_\\nu} \\geq T \\right \\},\n$$\nwith the threshold $T$ specified later. \n\nBy (\\ref{eq:deltanuproductbound}) and (\\ref{eq:Knuproductbound}) we see that $|J|$ is rather small:\n\\ben \\label{eq:AJbound}\n T^{|J|} \\leq \\prod_{\\nu \\in \\{0, 1 \\}^l}  \\frac{K_\\nu}{\\delta_\\nu} < \\lr{\\frac{20C_1}{c_1}}^{286 \\cdot l 2^{l}} \\log (\\frac{K}{\\delta})^{286 \\cdot 2^{l}} \\delta^{-21l}K.\n\\een\nSet $t := 2^l$, so it follows from (\\ref{eq:AJbound}) that for an appropriate constant $C_2$,\n$$\n|J| \\log T  \\leq C_2 l t.\n$$\nChoose\n\\begin{equation}\n \\log T := C_2\\gamma^{-1} l = \\frac{C_2 \\log \\log (K\/\\delta)}{\\gamma}.\n\\label{eq:Adefn}\n\\end{equation}\nThus \n\\begin{equation}\n\\frac{|J|}{t} \\leq \\frac{C_2 l}{\\log T} = \\gamma.\n\\label{eq:Jbound}\n\\end{equation}\n\nWe are finally ready to put everything together:\n\\begin{align*}\n\\psi(N,\\delta,K) &\\stackrel{\\eqref{eq:psibound}}{=}(Ck^2)^{2^l}\\prod_{\\nu \\in \\{0,1 \\}^l} \\min \\left\\{ (2k^2)^{\\left(\\frac{K_\\nu}{\\delta\\nu}\\right )^C}, N_\\nu \\right\\}. \n\\\\& \\leq (Ck^2)^{2^l}\\prod_{\\nu \\in I} (2k^2)^{T^C} \\prod_{\\nu \\in J} N_\\nu\n\\\\ & \\stackrel{\\eqref{eq:yetanotherNbound}}{\\leq} (C'k^2)^{tT^C}   \\left(\\frac{(20C_1)^{4000 l^3} \\left( \\log \\frac{K}{\\delta} \\right)^{4000 l^2}K^{22}}{\\delta^{4000 l}}N^{\\frac{1}{t}}\\right)^{|J|}\n\\\\& \\stackrel{\\eqref{eq:Jbound}}{\\leq} k^{ \\left( \\log \\frac{K}{\\delta} \\right)^{\\frac{C''}{\\gamma}}}N^{\\gamma}.\n\\end{align*}\n\n\n\\end{proof}\n\n\\section{A strong admissible pair} \\label{sec:strongpair}\n\nFinally, in this section we will use Lemma \\ref{lm:beteradmissiblepair} to get an even better pair of admissible functions. \n\\begin{Lemma} \\label{lm:finaldamissiblepairs}\nGiven $0 < \\tau, \\gamma < 1\/2$ there exist positive constants $\\alpha_i(\\tau, \\gamma,k), \\beta_i (\\tau, \\gamma,k), i= 1, 2, 3$ such that for all sufficiently large $N$, the pair\n\\ben\n  \\phi (N,\\delta,K):= K^{-\\alpha_1} \\delta^{\\alpha_2 \\log \\log N} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau} \\label{eq:bestphi} \\\\\n  \\psi (N, \\delta, K) :=   K^{\\beta_1} \\delta^{-\\beta_2 \\log \\log N} e^{-\\beta_3 (\\log \\log N)^2 } N^{\\gamma} \\label{eq:bestpsi}\n\\een\nis admissible. \n\n\\end{Lemma}\n\\begin{proof}\n\nThe strategy of the proof is as follows. We start with the already not-so-bad admissible pair given by Lemma \\ref{lm:beteradmissiblepair} and improve it by repeated application of Lemma \\ref{lm:InductionStepLemma}. \n\nLet $P_N[\\phi, \\psi]$ be the predicate that the pair $(\\phi, \\psi)$ given by (\\ref{eq:bestphi}) and (\\ref{eq:bestpsi}) is admissible in the sense of Definition \\ref{def:admissible_pairs} for all graphs of size at most $N$ and at least $N^{1\/2}$. \n\nWe are going to prove that \n\\begin{enumerate} \n   \\item The base case: $P_{N_0}[\\phi, \\psi]$ is true for some $N_0(\\tau, \\gamma)$.\n   \n   \\item The inductive step: $P_N[\\phi, \\psi] \\Rightarrow P_{N^{3\/2}}[\\phi, \\psi]$.\n\\end{enumerate}\nThe exponent $3\/2$ is of little importance here and is taken with much room to spare. Lemma \\ref{lm:finaldamissiblepairs} will then follow by induction, for all $N\\geq N_0$.\n\nIn order to prove (1) it suffices to find a fixed threshold $N_0(\\tau, \\gamma)$ such that the pair (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) is either trivial or worse than that given by Lemma \\ref{lm:beteradmissiblepair} if $N \\leq N_0$. One can achieve this by fine-tuning the constants $\\alpha_1, \\beta_1$, which we now explain. \n\n\n\n\n\n\n\n\n\nApply Lemma \\ref{lm:beteradmissiblepair} with $\\gamma= \\gamma \/4$ to obtain an admissible pair given by (\\ref{eq:betterphi}), (\\ref{eq:betterpsi}). We seek to choose $\\alpha_1, \\beta_1$ and  $N_0(\\delta, \\gamma)$ such that for each $N$ in the range $N_0^{1\/2} \\leq N \\leq N_0$\n\\ben\n\t\\left( \\frac{\\delta}{K}\\right)^{C(\\gamma) \\log \\log (K\/\\delta)} N \\geq K^{-\\alpha_1} \\delta^{\\alpha_2 \\log \\log N} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau} \\label{eq:bestphi2} \\\\\n \\min \\{N,\t\\exp\\lr{\\log k \\cdot \\log (K\/\\delta)^{C(\\gamma)\/\\gamma}} N^{\\gamma\/4} \\} \\leq K^{\\beta_1} \\delta^{-\\beta_2 \\log \\log N} e^{-\\beta_3 (\\log \\log N)^2 } N^{\\gamma} . \\label{eq:bestpsi2} \n\\een\nTo ensure (\\ref{eq:bestphi2}) holds it is sufficient to take $\\alpha_1 = \\frac{C(\\gamma)}{2}\\log \\log N_0$ with  $C(\\gamma) > 0$ from Lemma~\\ref{lm:beteradmissiblepair} and to take $\\alpha_2 = C_2 \\alpha_1$ and $\\alpha_3=C_3 \\alpha_1$ for some absolute constants $C_2,C_3 \\geq 1$. Indeed,\n\\begin{align*} K^{-\\alpha_1} \\delta^{\\alpha_2 \\log \\log N} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau} & \n\\leq \\left(\\frac{\\delta}{K} \\right)^{\\alpha_1} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau}\n\\\\& \\leq \\left(\\frac{\\delta}{K} \\right)^{C(\\gamma) \\log \\log N} e^{\\alpha_3 (\\log \\log N)^2 } N^{1 - \\tau}\n\\\\& \\leq \\left(\\frac{\\delta}{K} \\right)^{C(\\gamma) \\log \\log N}  N,\n\\end{align*}\nwhere the last inequality holds as long as we take $N_0$ sufficiently large (and thus also $N$ is sufficiently large). Inequality \\eqref{eq:bestphi2} then follows since the inequality $N \\geq \\frac{K}{\\delta}$ holds by definition of $N,\\delta$ and $K$.\n\n\nEnsuring (\\ref{eq:bestpsi2}) is more involved, as later on want to impose the further constraint $\\beta_3 > \\beta_2 > \\beta_1$.\nFor now, it suffices to guarantee that \n\\ben \\label{eq:Kboundexponencial}\n \\log k \\cdot \\log \\left( \\frac{K}{\\delta} \\right)^{\\frac{ C}{\\gamma}} < \\frac{\\gamma}{4} \\log N\n\\een\nand\n\\ben \\label{eq:B_3boundexponenial}\ne^{\\beta_3 (\\log \\log N)^2} < N^{\\frac{\\gamma}{2}}.\n\\een\nHowever, the bound (\\ref{eq:Kboundexponencial}) fails only if $K\/\\delta$ is rather large, namely\n$$\n\\frac{K}{\\delta} > e^{\\log^{c\\gamma} N}\n$$\nfor some $c(C, \\gamma, k) > 0$.  In this case it suffices to take $\\beta_1$ so large that \n\\[K^{\\beta_1} \\delta^{-\\beta_2 \\log \\log N} e^{-\\beta_3 (\\log \\log N)^2 } N^{\\gamma} > N\n\\]\nand thus (\\ref{eq:bestpsi2})  holds. To this end, we set\n$$\n\\beta_1 := (\\log N_0)^{1 - c\\gamma}\n$$\nand make the constraint that, say,\n$$\n\\beta_3, \\beta_2 < 10 \\beta_1  \\log \\log {N_0}.\n$$ \nMoreover, this constraint on $\\beta_3$ also ensures that \\eqref{eq:B_3boundexponenial} holds for $N$ sufficiently large.\n\nSumming up, we have found some fixed threshold $N_0(\\tau, \\gamma)$  at which (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) become admissible with fixed $\\alpha_1, \\beta_1$ and still some freedom to define the constants $\\alpha_2, \\beta_2, \\alpha_3$, and $\\beta_3$.\n\nWe now turn to part (2) of the induction scheme, the inductive step. Assuming that $N', N''$ are at the scale so that (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) are admissible with the data $(N', \\delta', K'); (N'', \\delta'', K'')$ we will show that  (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) are also admissible for the data $(N, \\delta, K)$ with $N \\approx N'N''$.\n\nAssuming $\\beta_1$ (or $N_0$) is large enough we may assume that \n\\ben \\label{eq:Kdeltasmall}\n\\frac{K}{\\delta} < N^{10^{-4}},\n\\een\nas otherwise (\\ref{eq:bestpsi})  $ > N$ which is trivially admissible.  \n\nWe need to estimate \n$$\n \\psi (N', \\delta', K') \\psi  (N'', \\delta'', K'') \n$$\nfrom above and \n$$ \n \\phi(N', \\delta', K') \\phi(N'', \\delta'', K''),\n$$\nfrom below in order to verify that (\\ref{eq:bestphi}), (\\ref{eq:bestpsi}) are admissible for $(N, \\delta, K)$. By (\\ref{eq:Kdeltasmall}), the constraints (\\ref{eq:parameterconstraints}) can be relaxed to\n\\ben\nN \\geq N'N'' &>& N\\left( \\frac{\\delta}{\\log N}\\right)^{40} > N^{99\/100} \\label{eq:N_1N_2simple} \\\\\nN' + N'' &<& N^{1\/2} \\left(\\frac{K}{\\delta} \\right)^{45} < N^{1\/2 + 1\/40}  \\label{eq:N_1plusN_2simple}\\\\\n\\delta' \\delta'' &>& \\frac{\\delta}{\\log^6 N} \\label{eq:deltadeltaprimesimple}\\\\\nK'K'' &<& \\delta^{-20} (\\log N)^{15}K \\label{eq:KprimeKprime} .\n\\een \n\nFrom (\\ref{eq:N_1N_2simple}) and (\\ref{eq:N_1plusN_2simple}) we have (with room to spare)\n\\beq\nN^{1\/2 - 1\/20} < N', N'' < N^{1\/2 + 1\/20}\n\\eeq\nand so assuming $N$ is large enough\n\\beq\n\\frac{99}{100} \\log \\log N < \\log \\log N',\\ \\log \\log N'' < \\log \\log N - \\log \\frac{20}{11}.\n\\eeq\nWith the constraints above, it suffices to verify (writing $l l$ for $\\log \\log $ as in \\cite{BC}) that\n\\ben \\label{eq:bestphisubstitution}\n(K'K'')^{-\\alpha_1} (\\delta')^{\\alpha_2 l l N'} (\\delta'')^{\\alpha_2 l l N''} e^{\\alpha_3[(l l N')^2 + (l l N'')^2]} (N' N'')^{1-\\tau}\n\\een\nis indeed always bounded below by (\\ref{eq:bestphi}).\nWe can bound (\\ref{eq:bestphisubstitution}) by\n\\ben\nK^{-\\alpha_1} \\delta^{\\alpha_2 l l N} e^{\\alpha_3 (l l N)^2} N^{1-\\tau} u \\cdot v\n\\een\nwhere\n\\ben\nu &=& (\\log N)^{-15\\alpha_1 - 6\\alpha_2 l l N - 40} e^{\\frac{9}{10}\\alpha_3 (l l N)^2} \\\\\nv &=& \\delta^{20\\alpha_1 - \\log \\frac{20}{11} \\alpha_2 + 40}.\n\\een\nFor suitable choices of $\\alpha_2, \\alpha_3 > \\alpha_1$ both $u, v > 1$ so  (\\ref{eq:bestphi}) is admissible. \n\nSimilarly for (\\ref{eq:bestpsi}) we have\n\\ben\n(K'K'')^{\\beta_1} (\\delta')^{-\\beta_2 l l N'} (\\delta'')^{-\\beta_2 l l N''} e^{-\\beta_3[(l l N')^2 + (l l N'')^2]} (N' N'')^{\\gamma} \\\\\n\t\t\t\t\t<K^{\\beta_1} \\delta^{-\\beta_2 l l N} e^{-\\beta_3 (l l N)^2} N^{\\gamma} u \\cdot v\n\\een\nwith\n\\ben\nu &=& (\\log N)^{15 \\beta_1 + 6\\beta_2 l l N} e^{-\\frac{9}{10}\\beta_3 (l l N)^2} \\\\\nv &=& \\delta^{-20\\beta_1 + \\log \\frac{20}{11} \\beta_2 }.\n\\een\nAgain, by taking suitable $\\beta_3 > \\beta_2 > \\beta_1$ we make $u, v < 1$ so (\\ref{eq:bestpsi}) is admissible. This closes the induction on scales argument and finishes the proof.\n\n\n\\end{proof}\n\n\\section{Concluding the proof of Theorem \\ref{thm:goodsubset} } \\label{sec:conclusion2}\nWe are finally ready to finish the proof of Theorem \\ref{thm:goodsubset}. Recall that the aim is to show that, given $0 < \\tau, \\gamma < 1\/2$, there are positive constants $C_1=C_1(\\tau,\\gamma,k)$ and $C_2=C_2(\\tau,\\gamma,k)$, such that for any $A \\subset \\mathbb Q$ with $|AA| \\leq K|A|$, there exists $A' \\subset A$ with $|A'| \\geq K^{-C_1}|A|^{1-\\tau}$, such that $A'$ is $K^{C_2}|A|^{\\gamma}$-separating.\n\nSince $|AA| \\leq K|A|$, after applying the prime evaluation map, we have $|\\mathcal P(A) +\\mathcal P(A)| \\leq K|\\mathcal P(A)|$. Fix $\\gamma'=\\gamma\/2$, $\\tau'=\\tau\/2$, and apply Lemma \\ref{lm:finaldamissiblepairs} for this choice of $\\gamma', \\tau'$, with the full graph $G= \\mathcal P(A) \\times \\mathcal P(A)$. It follows that there is a subgraph $G' \\subset G$ such\n\\[|G'| \\geq K^{-\\alpha_1} e^{\\alpha_3(\\log\\log |A|)^2} |A|^{2-2 \\tau'} \\geq K^{-\\alpha_1}  |A|^{2-2 \\tau'} \\]\nand such that for each $v \\in V(G)$ the $\\mathcal P$-preimages of $N_{G'}(v)$ is\n\\[K^{\\beta_1} e^{-\\beta_3(\\log\\log |A|)^2} |A|^{2\\gamma'} \\leq K^{\\beta_1} |A|^{2\\gamma'} \\]\nseparating.\\footnote{Note here that we have discarded the extra information coming from the terms of the form $e^{\\pm C (\\log\\log |A|)^2}$.} \n\nThen, by the pigeonhole principle, there is a vertex $v \\in V(G)$ such that $|N_{G'}(v)| \\geq |A|^{1-2 \\tau'}$. Write $A'=\\mathcal P^{-1}(N_{G'}(v))$ for the preimage of the neighbourhood of $v$. Then this is a subset of $A$ with the required properties.\n\n\\section{Further Applications} \\label{sec:applications}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:STinverse}] Recall that Theorem \\ref{thm:STinverse} is the following statement. For all $\\gamma \\geq 0$ there exists a constant $C=C(\\gamma)$ such that for any finite $A\\subset \\mathbb Q$ with $|AA| \\leq K|A|$ and any finite set $L$ of lines in the plane, $I(P,L)\\leq 3 |P| + |A|^{\\gamma}K^C|L|$, where $P= A \\times A$.\n\nFirst of all, observe that horizontal and vertical lines contribute a total of at most $2|P|$. This is because each point $p \\in P$ can belong to at most one horizontal and one vertical line. Similarly, lines through the origin contribute at most $|P| +|L|$ incidences, since each point aside from the origin belongs to at most one such line, and the origin itself may contribute $|L|$ incidences.\n\nIt remains to bound incidences with lines of the form $y=mx+c$, with $m,c \\neq 0$. Let $l_{m,c}$ denote the line with equation $y=mx+c$. Note that, if $m \\notin \\mathbb Q$ then $l_{m,c}$ contains at most one point from $P$. Indeed, suppose $l_{m,c}$ contains two distinct points $(x,y)$ and $(x',y')$ from $P$. In particular, since $A \\subset \\mathbb Q$, $x,y,x',y' \\in \\mathbb Q$. Then $l_{m,c}$ has direction $m= \\frac{y-y'}{x-x'}$. Therefore, lines $l_{m,c}$ with irrational slope $m$ contribute at most $|L|$ incidences.\n\nNext, suppose that $m \\in \\mathbb Q$ and $c \\notin \\mathbb Q$. Then $l_{m,c}$ does not contain any points from $P$, since if it did then we would have a solution to $y=mx+c$, but the left hand side is rational and the right hand side is irrational.\n\nIt remains to consider the case when $m,c \\in \\mathbb Q^*$. An application of Theorem \\ref{thm:BS_almost_subgroups} implies that $|l_{m,c} \\cap P| \\leq K^C|A|^{\\gamma}$. Therefore, these lines contribute a total of at most $|L|K^C|A|^{\\gamma}$ incidences.\n\nAdding together the contributions from these different types of lines completes the proof.\n\n\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:additivebasis}]\nRecall that Theorem \\ref{thm:additivebasis} states that, for any $\\gamma > 0$ there exists $C(\\gamma)$ such that\nfor an arbitrary $A \\subset \\mathbb{Q}$ with $|AA| = K|A|$ and $B, B' \\subset \\mathbb{Q}$,\n$$\nS := \\left|\\{(b, b') \\in B \\times B' : b + b' \\in A\\} \\right| \\leq 2|A|^\\gamma K^C \\min \\{|B|^{1\/2}|B'| + |B|, |B'|^{1\/2}|B| + |B'| \\}. \n$$\n\nWe will prove that \n\\begin{equation}\nS \\leq 2|A|^\\gamma K^C(|B'|^{1\/2}|B| + |B'|).\n\\label{eq:case1}\n\\end{equation}\nSince the roles of $B$ and $B'$ are interchangeable, \\eqref{eq:case1} also implies that $S \\leq 2|A|^\\gamma K^C(|B|^{1\/2}|B'| + |B|)$, and thus completes the proof.\n\nLet $\\gamma > 0$ and $C(\\gamma)$, given by Theorem~\\ref{thm:BS_almost_subgroups}, be fixed. Without loss of generality assume that $S \\geq 2|B'|$ as otherwise the claimed bound is trivial. \n\nFor each $b \\in B$ define \n$$\nS_b := \\{ b' \\in B' : b + b' \\in A\\},\n$$\nand similarly for $b' \\in B'$\n$$\nT_{b'} := \\{b \\in B: b' + b \\in A \\}.\n$$\nIt follows from Theorem~\\ref{thm:BS_almost_subgroups} that for $b_1,b_2 \\in B$ with $b_1 \\neq b_2$ \n$$\n|S_{b_1} \\cap S_{b_2}| \\leq |A|^\\gamma K^C\n$$\nsince each $x \\in S_{b_1} \\cap S_{b_2}$ gives a solution $(a, a') := (b_1 + x, b_2 + x)$ to\n$$\na - a' = b_1 - b_2\n$$\nwith $a, a' \\in A$.\n\nOn the other hand, by double-counting and the Cauchy-Schwarz inequality,\n$$\n\\sum_{b \\in B} |S_b| + \\sum_{b_1,b_2 \\in B : b_1 \\neq b_2} |S_{b_1} \\cap S_{b_2}| =\n\\sum_{b' \\in B'} |T_{b'}|^2 \\geq |B'|^{-1}(\\sum_{b' \\in B'} |T_{b'}|)^2 = |B'|^{-1}S^2.\n$$\nTherefore,\n$$\n\\sum_{b_1,b_2 \\in B : b_1 \\neq b_2} |S_{b_1} \\cap S_{b_2}| \\geq |B'|^{-1}S^2 - \\sum_{b \\in B} |S_b| = |B'|^{-1}S^2 - S \\geq \\frac{1}{2} |B'|^{-1}S^2\n$$\nby our assumption.\n\n\nThe left-hand side is at most $ |B|^2|A|^\\gamma K^C$, and so\n$$\n\tS \\leq (2|A|^\\gamma K^C)^{1\/2} |C|^{1\/2}|B'|,\n$$\nwhich completes the proof.\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:proddiff}]\nRecall that Theorem \\ref{thm:proddiff} states that for all $b$ there exists $k$ such that for all $A,B \\subset \\mathbb Q$ with $|B| \\geq 2$, $|(A+B)^{k}| \\geq |A|^b$.\n\nSince $|B| \\geq 2$, there exist two distinct elements $b_1, b_2 \\in B$. Apply Theorem \\ref{thm:mainmain} to conclude that for all $b$ there exists $k=k(b)$ with\n\\[ |(A+B)^k| \\geq  \\max \\{|(A+b_1)^k|, |((A+b_1)+(b_2-b_1))^{k} |\\} \\geq |A|^b.\\]\n\n\n\\end{proof}\n\n\\section*{Acknowledgements}\nOliver Roche-Newton was partially supported by the Austrian Science Fund FWF Project P 30405-N32. Dmitrii Zhelezov was supported by the Knut and Alice Wallenberg Foundation Program for Mathematics 2017. \n\nWe thank Brendan Murphy, Imre Ruzsa and Endre Szemer\\'edi for helpful conversations.\n\n\n\n\n\n\\AtEndEnvironment{thebibliography}{\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}