Weighted Diophantine approximation on manifolds

TL;DR

This paper proves a comprehensive weighted Khintchine-type theorem for nondegenerate manifolds, extending previous results and including a multiplicative convergence theorem, advancing the understanding of Diophantine approximation on manifolds.

Contribution

It establishes a weighted Khintchine-type theorem for all nondegenerate manifolds, addressing an open problem and extending prior work in the field.

Findings

Proves weighted simultaneous Khintchine-type theorem for nondegenerate manifolds.

Derives a multiplicative Khintchine-type convergence theorem for manifolds.

Extends previous results to a more general weighted setting.

Abstract

We establish a weighted simultaneous Khintchine-type theorem, both convergence and divergence, for all nondegenerate manifolds, which answers a problem posed in [Math. Ann., 337(4):769-796, 2007]. This extends the main results of [Acta Math., 231:1-30, 2023] and [Ann. of Math. (2), 175(1):187-235, 2012] in the weighted set-up. As a by-product of our method, we also obtain a multiplicative Khintchine-type convergence theorem for all nondegenerate manifolds, which is a simultaneous analogue of the celebrated result of Bernik, Kleinbock, and Margulis for dual approximation.