Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Chatzakos, Dimitrios, Dimakis, Panagiotis
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214716391