Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions

TL;DR

This paper establishes convergence results for Khintchine-Groshev type theorems on nondegenerate manifolds, extending classical metric number theory to geometric settings using lattice geometry techniques.

Contribution

It proves the convergence part of Khintchine-Groshev theorems for manifolds, introducing a novel approach combining metric number theory with lattice geometry.

Findings

Convergence results for standard Khintchine-Groshev theorem on manifolds

Extension to multiplicative versions of the theorem

New methods involving geometry of lattices in Euclidean spaces

Abstract

An analogue of the convergence part of the Khintchine-Groshev theorem, as well as its multiplicative version, is proved for nondegenerate smooth submanifolds in $\mathbb{R}^n$. The proof combines methods from metric number theory with a new approach involving the geometry of lattices in Euclidean spaces.