Large hyperbolic circles

TL;DR

This paper studies the distribution of large hyperbolic circles on compact hyperbolic surfaces, providing precise asymptotics, statistical limit theorems, and applications to the hyperbolic circle problem using spectral methods.

Contribution

It establishes new asymptotic formulas for hyperbolic circle equidistribution and derives statistical limit theorems, extending previous methods to a broader setting.

Findings

Precise asymptotics for hyperbolic circle equidistribution.

Statistical limit theorems with compactly supported distributions.

Applications to the hyperbolic circle problem.

Abstract

We consider circles of common centre and increasing radius on a compact hyperbolic surface and, more generally, on its unit tangent bundle. We establish a precise asymptotics for their rate of equidistribution. Our result holds for translates of any circle arc by arbitrary elements of $\text{SL}_2(\mathbb{R})$. Our proof relies on a spectral method pioneered by Ratner and subsequently developed by Burger in the study of geodesic and horocycle flows. We further derive statistical limit theorems, with compactly supported limiting distribution, for appropriately rescaled circle averages of sufficient regular observables. Finally, we discuss applications to the classical circle problem in the hyperbolic plane, following the approach of Duke-Rudnick-Sarnak and Eskin-McMullen.