A Central Limit Theorem for the Winding Number of Low-Lying Closed Geodesics
Number Theory
2026-01-28 v2 Dynamical Systems
Abstract
We show that the winding of low-lying closed geodesics on the modular surface has a Gaussian limiting distribution when normalized by any standard notion of length, in contrast to the Cauchy distribution arising when allowing arbitrarily deep excursions into the cusp. In addition, we prove a Berry-Esseen bound and a local limit theorem.
Cite
@article{arxiv.2507.23706,
title = {A Central Limit Theorem for the Winding Number of Low-Lying Closed Geodesics},
author = {Elias Dubno},
journal= {arXiv preprint arXiv:2507.23706},
year = {2026}
}
Comments
Revised version. Added Berry-Esseen bounds and a local limit theorem; the comparison theorem was slightly reorganized