Integer points on spheres and their orthogonal grids

TL;DR

This paper studies the distribution of primitive vectors on large spheres in higher dimensions, focusing on their directions and orthogonal lattice shapes, using unipotent dynamics to prove equidistribution results.

Contribution

It refines previous equidistribution results by analyzing the joint distribution of vector directions and orthogonal lattice shapes in dimensions greater than 3, 4, and 5.

Findings

Established equidistribution in dimensions d>5.

Proved equidistribution in dimensions d=4,5 under mild conditions.

Extended understanding of lattice point distribution on spheres.

Abstract

The set of primitive vectors on large spheres in the euclidean space of dimension d>2 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concerning the direction of the vector together with the shape of the lattice in its orthogonal complement. Using unipotent dynamics we obtained the desired equidistribution result in dimension d>5 and in dimension d=4,5 under a mild congruence condition on the square of the radius. The case of d=3 is considered in a separate paper.