Twisted Diophantine approximation on manifolds

TL;DR

This paper explores twisted Diophantine approximation on manifolds, introducing new concepts of twisted Khintchine type for convergence/divergence and analyzing approximation properties on nondegenerate manifolds.

Contribution

It defines twisted Khintchine type for manifolds and provides conditions for nondegenerate analytic manifolds to exhibit this behavior.

Findings

Identifies conditions for twisted Khintchine type on manifolds

Analyzes intersection properties of approximation sets with manifolds

Establishes new framework for twisted Diophantine approximation

Abstract

In twisted Diophantine approximation, for a fixed $m\times n$ matrix $\boldsymbol\alpha$ one is interested in sets of vectors $\boldsymbol\beta\in\mathbb R^m$ such that the system of affine forms $\mathbb R^n \ni \mathbf q \mapsto \boldsymbol\alpha\mathbf q + \boldsymbol\beta \in \mathbb R^m$ satisfies some given Diophantine condition. In this paper we introduce the notion of manifolds which are of $\boldsymbol\alpha$-twisted Khintchine type for convergence or divergence. We provide sufficient conditions under which nondegenerate analytic manifolds exhibit this twisted Khintchine-type behaviour. Furthermore, we investigate the intersection properties of the sets of $\boldsymbol\alpha$-twisted badly approximable and well approximable vectors with nondegenerate manifolds.