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

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 $\mathbb{R}^{M}$.