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

TL;DR

The paper analyzes the winding number $\\Psi(C)$ of closed geodesics on the modular surface under a cusp-restricted regime, showing a Gaussian limiting distribution when the winding is scaled by the square root of the period length for $A$-low-lying geodesics. It introduces the $A$-low-lying framework via bounded continued-fraction partial quotients and proves a central limit theorem with variance $\\sigma_p^2=(A^2-1)/12$, complemented by a Berry-Esseen bound and a local limit theorem. A key contribution is a Comparison Theorem establishing that standard length notions (geometric, word, maximal) are asymptotically equivalent to period length in this regime, ensuring the Gaussian limit is robust to the chosen length normalization. The results also discuss a cusp-driven transition to the Cauchy law if the truncation parameter $A$ grows with the period bound, and provide explicit variance formulas for different normalizations, highlighting the cusp’s role in limiting behavior.

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.

Figures (3)

• Figure 1: Illustration of Main Results for $A=5$. Empirical distributions are shown in blue and the corresponding limiting Gaussians in dashed red.
• Figure 2: The closed geodesic $[\overline{3,2,3,4}]$.
• Figure 3: Proof idea for Lemma \ref{['lem:lastlem']}

Theorems & Definitions (43)

• Remark 1.0
• Definition 1.0
• Theorem 1.0: Main Central Limit Theorem
• Theorem 1.0: Berry--Esseen
• Theorem 1.0: Local Limit Theorem
• Definition 1.0
• Theorem 1.0: Comparison Theorem
• Remark 1.0
• Theorem 1.0: CLT with different lengths
• Remark 1.0