Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory

TL;DR

This paper advances the convergence theory in Diophantine approximation on manifolds, showing that manifolds of sufficient dimension and non-degeneracy are of Khintchine type for convergence, based on improved bounds for rational point distribution.

Contribution

It establishes the convergence theory for inhomogeneous Diophantine approximation on manifolds of dimension greater than (n+1)/2, with optimal bounds on rational point distribution.

Findings

Manifolds of dimension > (n+1)/2 are of Khintchine type for convergence.

Develops the best possible bounds for rational points near manifolds.

Extends convergence results to inhomogeneous approximation scenarios.

Abstract

In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $M$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.