Twisted Diophantine approximation on manifolds

Abstract

In twisted Diophantine approximation, for a fixed $m\times n$ matrix $\hat{\mathbf{I}}\pm$ one is interested in sets of vectors ${\hat{\mathbf{I}}}^2\in {\mathbb{R}}^m$ such that the system of affine forms ${\mathbb{R}}^n\ni \mathbf{q}\mapsto \hat{\mathbf{I}}\pm \mathbf{q}+{\hat{\mathbf{I}}}^2\in {\mathbb{R}}^m$ satisfies some given Diophantine condition. In this paper we introduce the notion of manifolds which are of $\hat{\mathbf{I}}\pm$-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 $\hat{\mathbf{I}}\pm$-twisted badly approximable and well approximable vectors with nondegenerate manifolds.

32 pages