Lattice points in spherical segments

TL;DR

This paper investigates the distribution of lattice points within spherical segments in higher dimensions, providing an upper bound based on geometric and Diophantine approximation techniques.

Contribution

It introduces a new upper bound for lattice points in spherical segments depending solely on radius and opening angle, utilizing slicing by rational hyperplanes.

Findings

Established an upper bound for lattice points in spherical segments

Applied Diophantine approximation to optimize rational directions

Demonstrated the method's effectiveness in higher dimensions

Abstract

We study lattice points in d-dimensional spheres, and count their number in thin spherical segments. We found an upper bound depending only on the radius of the sphere and opening angle of the segment. To obtain this bound we slice the segment by hyperplanes of rational direction, and then cover an arbitrary segment with one having rational direction. Diophantine approximation can be used to obtain the best rational direction possible.