Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals

TL;DR

This paper introduces a new method combining non-divergence estimates and Fourier analysis to study rational points near manifolds, leading to novel bounds, asymptotics, and measure results for Diophantine approximation.

Contribution

The authors develop a novel approach that avoids traditional geometric methods, providing the first asymptotic formulas and improved bounds for rational points near non-analytic manifolds.

Findings

Established strong lower bounds on the number of rational points near manifolds.

Derived asymptotic formulas for rational point counts on manifolds.

Improved bounds and measure results for well-approximable points.

Abstract

Let $\mathcal{M}\subset \mathbb{R}^n$ be a compact and sufficiently smooth manifold of dimension $d$. Suppose $\mathcal{M}$ is nowhere completely flat. Let $N_{\mathcal{M}}(\delta,Q)$ denote the number of rational vectors $\mathbf{a}/q$ within a distance of $\delta/q$ from $\mathcal{M}$ so that $q \in [Q,2Q)$. We develop a novel method to analyse $N_{\mathcal{M}}(\delta,Q)$. The salient feature of our technique is the combination of powerful quantitative non-divergence estimates, in a form due to Bernik, Kleinbock, and Margulis, with Fourier analytic tools. The second ingredient enables us to eschew the Dani correspondence and an explicit use of the geometry of numbers. We employ this new method to address in a strong sense a problem of Beresnevich regarding lower bounds on $N_{\mathcal{M}}(\delta,Q)$ for non-analytic manifolds. Additionally, we obtain asymptotic formulae which are the first of their kind for such a general class of manifolds. As a by-product, we improve upon upper bounds on $N_{\mathcal{M}}(\delta,Q)$ from a recent breakthrough of Beresnevich and Yang and recover their convergence Khintchine type theorem for arbitrary nondegenerate submanifolds. Moreover, we obtain new Hausdorff dimension and measure refinements for the set of well-approximable points for a range of Diophantine exponents close to $1/n$.