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 $\mathcal{M}$ of $\mathbb{R}^M$ with a certain curvature condition. Our result generalises earlier work of Huang for hypersurfaces [J.-J. Huang, The density of rational points near hypersurfaces, Duke Math. J. 169 (2020), 2045--2077.], as our curvature condition reduces to Gaussian curvature being bounded away from $0$ when $M - dim \mathcal{M} = 1$. An interesting feature of our result is that the asymptotic formula holds beyond the conjectured range of the distance to $\mathcal{M}$. Furthermore, we obtain an upper bound for the number of rational points on $\mathcal{M}$ with additional power saving to the bound in the analogue of Serre's dimension growth conjecture for compact submanifolds of $\mathbb{R}^M$ when $M - dim \mathcal{M} > 1$.