Schmidt's game, fractals, and orbits of toral endomorphisms

Given an integer nonsingular $n\times n$ matrix $M$ and a point $y \in $\mathbb{R}$^n/$\mathbb{Z}$^n$,considertheset$\tilde E(M,y)$ofvectors$x\in $\mathbb{R}$^n$suchthat$y$isnotalimitpointofthesequence${M^k x \mod $\mathbb{Z}$^n: k\in$\mathbb{N}$}$\mathrm{.}S\mathrm{.}G\mathrm{.}Danishowedin1988thatwhenever$M$ is semisimple and $y\in {\mathbb{Q}}^n\mathrm{/}{\mathbb{Z}}^n$, the set $\tilde{E}(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary $y\in {\mathbb{R}}^n\mathrm{/}{\mathbb{Z}}^n$ and integer nonsingular $M$, and in fact replacing the sequence of powers of $M$ by any lacunary sequence of (not necessarily integer) $m\times n$ matrices. Furthermore, we show that sets of the form $\tilde{E}(M,y)$ and their generalizations always intersect with `sufficiently regular’ fractal subsets of ${\mathbb{R}}^n$. As an application we give an alternative proof of a recent result of Einsiedler and Tseng on badly approximable systems of affine forms.