  Given an integer nonsingular $n\times n$ matrix $M$ and a point $y \in
\mathbb{R}^n/\mathbb{Z}^n$, consider the set $\tilde E(M,y)$ of vectors $x\in
\mathbb{R}^n$ such that $y$ is not a limit point of the sequence $\{M^k x \mod
\mathbb{Z}^n: k\in\mathbb{N}\}$. S.G. Dani showed in 1988 that whenever $M$ is
semisimple and $y \in \mathbb{Q}^n/\mathbb{Z}^n$, the set $\tilde E(M,y)$ has
full Hausdorff dimension. In this paper we strengthen this result, extending it
to arbitrary $y \in \mathbb{R}^n/\mathbb{Z}^n$ 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.
