Counting Rational Points In Non-Isotropic Neighborhoods of Manifolds

TL;DR

This paper studies the distribution of rational points near curved manifolds in non-isotropic neighborhoods, providing bounds and asymptotics that extend previous results and have applications in Diophantine approximation.

Contribution

It establishes new upper bounds and asymptotic formulas for rational points near manifolds under strong curvature conditions, generalizing prior work and applying to Hausdorff dimension estimates.

Findings

Derived upper bounds for rational points in non-isotropic neighborhoods.

Obtained asymptotic formulas beyond previous distance ranges.

Established new bounds for rational points on manifolds, surpassing existing conjectures.

Abstract

In this manuscript, we initiate the study of the number of rational points with bounded denominators, contained in a non-isotropic $\delta_1\times\ldots\times \delta_R$ neighborhood of a compact submanifold $\mathcal{M}$ of codimension $R$ in $\mathbb{R}^{M}$. We establish an upper bound for this counting function which holds when $\mathcal{M}$ satisfies a strong curvature condition, first introduced by Schindler-Yamagishi in \cite{schindler2022density}. Further, even in the isotropic case when $\delta_1=\ldots=\delta_R=\delta$, we obtain an asymptotic formula which holds beyond the range of distance to $\mathcal{M}$ established in \cite{schindler2022density}. Our result is also a generalization of the work of J.J. Huang \cite{huangduke} for hypersurfaces. As an application, we establish for the first time an upper bound for the Hausdorff dimension of the set of weighted simultaneously well approximable points on a manifold $\mathcal{M}$ satisfying the strong curvature condition, which agrees with the lower bound obtained by Allen-Wang in \cite{allen2022note}. Moreover, for $R>1$, we obtain a new upper bound for the number of rational points \textit{on} $\mathcal{M}$, which goes beyond the bound in an analogue of Serre's dimension growth conjecture for submanifolds of $\mathbb{R}^M$ .