Khintchine's theorem and Diophantine approximation on manifolds

TL;DR

This paper proves a Khintchine-type theorem for nondegenerate submanifolds of Euclidean space, advancing the understanding of rational approximation on manifolds and determining the Hausdorff dimension of well-approximable points.

Contribution

It introduces a new approach combining geometric and dynamical methods to resolve a longstanding problem in Diophantine approximation on manifolds.

Findings

Established a convergence Khintchine theorem for nondegenerate submanifolds.

Determined the Hausdorff dimension of τ-well approximable points on manifolds.

Provided explicit bounds for rational points near manifolds.

Abstract

In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary nondegenerate submanifolds of $\mathbb{R}^n$, which resolves a longstanding problem in the theory of Diophantine approximation. Furthermore, we refine this result using Hausdorff $s$-measures and consequently obtain the exact value of the Hausdorff dimension of $\tau$-well approximable points lying on any nondegenerate submanifold for a range of Diophantine exponents $\tau$ close to $1/n$. Our approach uses geometric and dynamical ideas together with a new technique of `generic and special parts'. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near the generic part of a non-degenerate manifold. In turn, we give an explicit exponentially small bound for the measure of the special part of the manifold. The latter uses a result of Bernik, Kleinbock and Margulis.