Density of rational points near/on compact manifolds with certain curvature conditions

Abstract

In this article we establish an asymptotic formula for the number of rational points, with bounded denominators, within a given distance to a compact submanifold M of RM with a certain curvature condition. Our result generalises earlier work of Huang for hypersurfaces (J.-J. Huang (2020) [18]), as our curvature condition reduces to Gaussian curvature being bounded away from 0 when M−dim⁡M=1. An interesting feature of our result is that the asymptotic formula holds beyond the conjectured range of the distance to M. Furthermore, we obtain an upper bound for the number of rational points on M with additional power saving to the bound in the analogue of Serre's dimension growth conjecture for compact submanifolds of RM when M−dim⁡M>1.