Rational Approximation on Spheres

TL;DR

This paper investigates how densely rational points are distributed on spheres, establishing approximation theorems that extend classical results and improve upon recent related work.

Contribution

It provides new Dirichlet, Khintchine, and badly approximable point theorems for rational approximation on spheres, enhancing prior understanding.

Findings

Every point on the sphere is sufficiently approximable by rational points.

Existence of badly approximable points on the sphere.

Measure of approximable points depends on convergence/divergence of a specific sum.

Abstract

We quantify the density of rational points in the unit sphere $S^n$, proving analogues of the classical theorems on the embedding of $\q^n$ into $\r^n$. Specifically, we prove a Dirichlet theorem stating that every point $\alpha \in S^n$ is sufficiently approximable, the optimality of this approximation via the existence of badly approximable points, and a Khintchine theorem showing that the Lebesgue measure of approximable points is either zero or full depending on the convergence or divergence of a certain sum. These results complement and improve on previous results, particularly recent theorems of Ghosh, Gorodnik and Nevo.