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arXiv · 2609.33829

Square root cancellation in the hyperbolic lattice counting problem over cocompact groups

Dimitrios Chatzakos·Panagiotis Dimakis
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Abstract

For cocompact Fuchsian groups $Γ\leq \hbox{PSL}(2, \mathbb{R})$ we improve Selberg's bound for the error term of the counting function in the hyperbolic lattice counting problem, achieving an essentially optimal upper bound for the error term \begin{equation*} E(X;z, w) = O(X^{1/2+\varepsilon}) \end{equation*} for almost every pair of points $z,w$. We extend this pointwise result for cocompact lattices acting on the $n$-dimensional real hyperbolic space $\mathbb{H}^n$. Moreover, in the $2$-dimensional case we study the second moment of the error term of the counting function. We improve the upper bounds of Chamizo and Cherubini for almost all pairs $z,w$, but we disprove a conjecture of Phillips and Rudnick for the case of cocompact Fuchsian groups.

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BibTeXRIS

Dimitrios Chatzakos, Panagiotis Dimakis. 2026-09-27. Square root cancellation in the hyperbolic lattice counting problem over cocompact groups. https://arxiv.org/abs/2609.33829

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