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

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.

arXiv commentFinal version. To appear in the special issue of the Bulletin of the Iranian Mathematical Society (BIMS) in honor of Freydoon Shahidi's 70th birthday

References in corpus (2)

• Linnik's ergodic method and the distribution of integer points on spheres
• Arithmetic and equidistribution of measures on the sphere

Cited by in corpus (4)

• Nodal intersections for random waves against a segment on the 3-dimensional torus
• Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics
• CLT for Planck scale mass distribution of toral Laplace eigenfunctions
• Random eigenfunctions on flat tori: universality for the number of intersections