A Central Limit Theorem for the Winding Number of Low-Lying Closed Geodesics

TL;DR

This paper proves that the winding number of low-lying closed geodesics on the modular surface follows a Gaussian distribution when normalized, contrasting with the Cauchy distribution for deeper excursions, and provides quantitative bounds.

Contribution

It establishes a central limit theorem for the winding number of low-lying geodesics, including Berry-Esseen bounds and a local limit theorem, highlighting a new probabilistic behavior.

Findings

Winding number of low-lying geodesics is Gaussian distributed.

Contrast with Cauchy distribution for deep excursions.

Provides Berry-Esseen bounds and local limit theorem.

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.