Local statistics of lattice points on the sphere

TL;DR

This paper surveys recent work on the local statistical properties of lattice points on the sphere, exploring beyond uniform distribution to understand finer-scale randomness and geometric configurations.

Contribution

It provides new insights into the local statistics of lattice points on the sphere, including electrostatic energy and spacing, extending understanding beyond classical uniform distribution results.

Findings

Evidence of randomness in local point configurations

Analysis of point pair statistics and nearest neighbor distributions

Comparison with random point models

Abstract

A celebrated result of Legendre and Gauss determines which integers can be represented as a sum of three squares, and for those it is typically the case that there are many ways of doing so. These different representations give collections of points on the unit sphere, and a fundamental result, conjectured by Linnik, is that under a simple condition these become uniformly distributed on the sphere. In this note we survey some of our recent work, which explores what happens beyond uniform distribution, giving evidence to randomness on smaller scales. We treat the electrostatic energy, local statistics such as the point pair statistic (Ripley's function), nearest neighbour statistics, minimum spacing and covering radius. We briefly discuss the situation in other dimensions, which is very different. In an appendix we compute the corresponding quantities for random points