Rational Points in Hyperbolic Regions and Multiplicative Diophantine Approximation on Manifolds

TL;DR

This paper proves the convergence of multiplicative Diophantine approximation on all non-degenerate manifolds and certain affine subspaces, answering longstanding questions and strengthening previous results in the field.

Contribution

It establishes the convergence theory for all non-degenerate smooth manifolds and affine subspaces under optimal Diophantine conditions, resolving open problems.

Findings

Confirmed convergence for all non-degenerate manifolds

Settled convergence for affine subspaces with generic Diophantine conditions

Sharpened extremality results for manifolds and affine subspaces

Abstract

We establish the convergence theory of multiplicative Diophantine approximation for all non-degenerate, smooth manifolds. We also settle said convergence theory for all affine subspaces satisfying a highly generic and essentially optimal Diophantine condition. This answers a question of Beresnevich and Velani from 2005, while simultaneously sharpening results of Kleinbock and Margulis on the strong extremality of non-degenerate manifolds, and of Kleinbock on the strong extremality of affine subspaces.