Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

TL;DR

This paper provides asymptotic estimates for the distribution of lattice points, Heegner points, and closed geodesics, partially resolving conjectures and revealing different behaviors in variance for specific subsequences.

Contribution

It introduces new variance estimates for lattice points and geodesics, addressing longstanding conjectures and highlighting differences in asymptotic behavior for special subsequences.

Findings

Variance estimates for lattice points in thin annuli and balls.

Partial resolution of Bourgain-Rudnick and Linnik conjectures.

Distinct asymptotic behaviors for squarefree integers and subsequences.

Abstract

We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a "microsquare", a conjecture of Bourgain and Rudnick on the number of lattice points lying in small balls on the surface of the sphere, the covering radius of the sphere, and the distribution of lattice points in almost all thin regions lying on the surface of the sphere. Finally, we show that for a density $1$ subsequence of squarefree integers, the variance exhibits a different asymptotic behaviour for balls of volume $(\log n)^{-\delta}$ with $0 < \delta < \tfrac {1}{16}$. We also obtain analogous results for Heegner points and closed geodesics. Interestingly, we are able to prove some slightly stronger results for closed geodesics than for Heegner points or lattice points on the surface of the sphere. A crucial observation that underpins our proof is the different behaviour of weights functions for annuli and for balls.