On the number of rational points close to a compact manifold under a less restrictive curvature condition

TL;DR

This paper derives an asymptotic formula for counting rational points near a compact manifold under a relaxed curvature condition, extending previous results and addressing conjectures in Diophantine approximation.

Contribution

It introduces a less restrictive curvature condition for counting rational points near manifolds, broadening the scope of previous geometric Diophantine results.

Findings

Established an asymptotic formula for rational points near manifolds

Generalized earlier curvature conditions in Diophantine approximation

Connected results to conjectures by Huang and Serre

Abstract

Let $\mathscr{M}$ be a compact submanifold of $\mathbb{R}^{M}$. In this article we establish an asymptotic formula for the number of rational points within a given distance to $\mathscr{M}$ and with bounded denominators under the assumption that $\mathscr{M}$ fulfills a certain curvature condition. Our result generalizes earlier work from Schindler and Yamagishi, as our curvature condition is a relaxation of that used by them. We are able to recover a similar result concerning a conjecture by Huang and a slightly weaker analogue of Serre's dimension growth conjecture for compact submanifolds of $\mathbb{R}^{M}$.