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Quantitative Equidistribution on Hyperbolic Surfaces and Arithmetic Applications

Peter Humphries

TL;DR

The paper develops a Berry–Esseen–type inequality for the $1$-Wasserstein distance on finite-area hyperbolic surfaces, expressing the bound in terms of spectral data from Maass cusp forms and Eisenstein series via an inverse Selberg–Harish-Chandra transform. It treats both cocompact and noncocompact lattices, employing a smoothing kernel and automorphic kernel to translate Lipschitz-test-function deviations into spectral sums, with cuspidal tightness ensuring control of Eisenstein contributions. The authors then apply the inequality to arithmetic equidistribution problems on $ ext{SL}_2(\\mathbb{Z})\backslash\mathbb{H}$: Duke’s equidistribution results for Heegner points and closed geodesics, and conditional (under GLH) mass equidistribution of Hecke–Maass cusp forms, recovering and sharpening rates via moments of $L$-functions. The framework connects optimal transport to deep automorphic–spectral tools (Waldspurger, Watson–Ichino) and yields explicit decay rates in terms of discriminants or spectral parameters, illustrating a powerful method to quantify arithmetic equidistribution. The work thus bridges probabilistic distance measures with analytic number theory in a way that invites further conditional improvements and extensions to related automorphic settings.

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.

Quantitative Equidistribution on Hyperbolic Surfaces and Arithmetic Applications

TL;DR

The paper develops a Berry–Esseen–type inequality for the -Wasserstein distance on finite-area hyperbolic surfaces, expressing the bound in terms of spectral data from Maass cusp forms and Eisenstein series via an inverse Selberg–Harish-Chandra transform. It treats both cocompact and noncocompact lattices, employing a smoothing kernel and automorphic kernel to translate Lipschitz-test-function deviations into spectral sums, with cuspidal tightness ensuring control of Eisenstein contributions. The authors then apply the inequality to arithmetic equidistribution problems on : Duke’s equidistribution results for Heegner points and closed geodesics, and conditional (under GLH) mass equidistribution of Hecke–Maass cusp forms, recovering and sharpening rates via moments of -functions. The framework connects optimal transport to deep automorphic–spectral tools (Waldspurger, Watson–Ichino) and yields explicit decay rates in terms of discriminants or spectral parameters, illustrating a powerful method to quantify arithmetic equidistribution. The work thus bridges probabilistic distance measures with analytic number theory in a way that invites further conditional improvements and extensions to related automorphic settings.

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 , 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.

Paper Structure

This paper contains 13 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. The Wasserstein Distance and the Berry--Esseen Inequality
  3. A Berry--Esseen Inequality for Finite Area Hyperbolic Surfaces
  4. The Cocompact Case
  5. The Noncocompact Case
  6. Arithmetic Applications
  7. Duke's Theorem on the Equidistribution of Heegner Points and of Closed Geodesics
  8. Mass Equidistribution of Hecke--Maaß Cusp Forms
  9. Tools
  10. Proofs of Theorems \ref{['thm:compact']} and \ref{['thm:noncompact']}
  11. Arithmetic Applications
  12. Proof of Theorem \ref{['thm:Duke']}
  13. Proof of Theorem \ref{['thm:MaassQUE']}

Key Result

Theorem 1.1

Let $\nu_1,\nu_2$ be Borel probability measures on $\mathbb{R}$. For $T \geq 1$, we have that

Theorems & Definitions (26)

  • Theorem 1.1: Berry--Esseen inequality Bob16
  • Theorem 1.2: Bobkov--Ledoux BL21, Borda Bor21a (see also KU25)
  • Theorem 1.5
  • Remark 1.7
  • Definition 1.8
  • Example 1.9
  • Theorem 1.10
  • Theorem 1.14: Duke Duk88 (see also ELMV12)
  • Theorem 1.19: Lindenstrauss--Soundararajan Lin06Sou10b
  • Theorem 1.20
  • ...and 16 more