Lattice points in spherical segments

2019/12/22 by Martin Ortiz Ramirez, Ramirez, Martin Ortiz
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1912.10462

openalex publication_date 2019/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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