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

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.

Revised version. Added Berry-Esseen bounds and a local limit theorem; the comparison theorem was slightly reorganized

References in corpus (3)

• The Manin-Drinfeld theorem and the rationality of Rademacher symbols
• Windings of prime geodesics
• A Lyapunov exponent attached to modular functions