Work

An Ω-result for the counting of geodesic segments in the hyperbolic plane

Year: 2025

Type: article

Abstract: Let Γ \Gamma be a cocompact Fuchsian group, and l l a fixed closed geodesic. We study the counting of those images of l l under the action of Γ \Gamma that have a distance from l l less than or equal to R R . We prove an Ω \Omega -result for the error term in the asymptotic expansion of the counting function. More specifically, we prove that, for every δ > 0 \delta >0 , the error term is equal to Ω δ ( X 1 / 2 ( log ⁡ log ⁡ X ) 1 / 4 − δ ) \Omega _{\delta }\left (X^{1/2}\left (\log {\log {X}}\right )^{1/4-\delta } \right ) , where X = cosh ⁡ R X=\cosh {R} .

Source: Proceedings of the American Mathematical Society

DOI: 10.1090/proc/17378

Author: Marios Voskou

Institutions: HUN-REN Alfréd Rényi Institute of Mathematics, Max Planck Institute for Mathematics

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: University College London, Vetenskapsrådet, Engineering and Physical Sciences Research Council

Awards: Maths DTP 2020 University College London, Institut Mittag-Leffler, G4317470034