Spatial statistics for lattice points on the sphere I: Individual results

TL;DR

This paper investigates the spatial distribution of lattice points on the sphere derived from sums of three squares, analyzing various statistical measures and their dependence on the Generalized Riemann Hypothesis.

Contribution

It introduces new statistical analyses of lattice points on the sphere and explores their properties under number-theoretic assumptions, advancing understanding of their distribution.

Findings

Electrostatic potential and Ripley's function are characterized for these point sets.

Variance of points in spherical caps is quantified, revealing distribution patterns.

Some results depend on the validity of the Generalized Riemann Hypothesis.

Abstract

We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares. We examine several statistics of these point sets, such as the electrostatic potential, Ripley's function, the variance of the number of points in random spherical caps, and the covering radius. Some of the results are conditional on the Generalized Riemann Hypothesis.