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arXiv · 2507.23706

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

Elias Dubno
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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.

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Elias Dubno. 2025-07-31. A Central Limit Theorem for the Winding Number of Low-Lying Closed Geodesics. https://doi.org/10.1007/s10711-026-01089-8

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