Quantitative Equidistribution on Hyperbolic Surfaces and Arithmetic Applications

TL;DR

This paper establishes a Berry-Esseen type inequality for Wasserstein distance on hyperbolic surfaces, linking equidistribution measures to Weyl sums, with applications to classical problems in number theory and quantum chaos.

Contribution

It introduces a novel inequality controlling Wasserstein distance via Weyl sums on hyperbolic surfaces, enabling new bounds in equidistribution problems.

Findings

Proved a Berry-Esseen inequality analogue for Wasserstein distance.

Derived upper bounds for equidistribution measures on the modular surface.

Connected equidistribution results to Weyl sums and automorphic forms.

Abstract

The Wasserstein distance quantifies the distance between two probability measures on a metric space. We prove an analogue of the Berry-Esseen inequality for the Wasserstein distance on a finite area hyperbolic surface. This inequality controls the Wasserstein distance via an average of Weyl sums, which are integrals of Maass cusp forms and Eisenstein series with respect to these probability measures. As applications, we prove upper bounds for the Wasserstein distance for some equidistribution problems on the modular surface $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$, namely Duke's theorems on the equidistribution of Heegner points and of closed geodesics and Watson's theorem on the mass equidistribution of Hecke-Maass cusp forms conditionally under the assumption of the generalised Lindelof hypothesis.