PreprintCC BY 4.0arXiv
Square root cancellation in the hyperbolic lattice counting problem over cocompact groups
Abstract
For cocompact Fuchsian groups Γ≤ PSL(2, 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 E(X;z, w) = O(X1/2+ε) for almost every pair of points z,w. We extend this pointwise result for cocompact lattices acting on the n-dimensional real hyperbolic space ℍn. 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.
Related
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- C1-rigidity of ℍp,q-convex cocompact representations of Gromov hyperbolic groups
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BibTeX
@misc{chatzakos2026square,
title = {{Square root cancellation in the hyperbolic lattice counting problem over cocompact groups}},
author = {Dimitrios Chatzakos and Panagiotis Dimakis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33829},
eprint = {2609.33829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33829},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Square root cancellation in the hyperbolic lattice counting problem over cocompact groups
AU - Dimitrios Chatzakos
AU - Panagiotis Dimakis
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33829
UR - https://arxiv.org/abs/2609.33829
AB - For cocompact Fuchsian groups Γ≤ PSL(2, 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 E(X;z, w) = O(X^1/2+ε) for almost every pair of points z,w. We extend this pointwise result for cocompact lattices acting on the n-dimensional real hyperbolic space 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.
ER -