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<p>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.</p>

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A central limit theorem for the winding number of low-lying closed geodesics

  • 期刊论文

  • 发表日期：

    2026年5月26日

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  • Elias Dubno

摘要

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.