Rational points near manifolds and metric Diophantine approximation

TL;DR

This paper proves divergence theorems in metric Diophantine approximation for analytic non-degenerate manifolds of any dimension, establishing the distribution of rational points near these manifolds.

Contribution

It introduces sharp lower bounds for rational points near non-degenerate manifolds in higher dimensions, advancing the understanding of Diophantine approximation on manifolds.

Findings

Established Khintchine and Jarnik divergence theorems for arbitrary analytic manifolds

Derived sharp lower bounds for the number of rational points near manifolds in dimensions n>2

Showed rational points are uniformly distributed near non-degenerate manifolds

Abstract

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindzuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarnik type divergence theorems are established for arbitrary analytic non-degenerate manifolds regardless of their dimension. The key to establishing these results is the study of the distribution of rational points near manifolds -- a very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near non-degenerate manifolds in dimensions n>2 and show that they are ubiquitous (that is uniformly distributed).