Twisted Diophantine approximation on manifolds

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