Given b > 1 and y \in \mathbb{R}/\mathbb{Z}$, we consider the set of $x\in
\mathbb{R}$ such that $y$ is not a limit point of the sequence $\{b^n x \bmod
1: n\in\N\}$. Such sets are known to have full Hausdorff dimension, and in many
cases have been shown to have a stronger property of being winning in the sense
of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets
and their bi-Lipschitz images must intersect with `sufficiently regular'
fractals $K\subset \mathbb{R}$ (that is, supporting measures $\mu$ satisfying
certain decay conditions). Furthermore, the intersection has full dimension in
$K$ if $\mu$ satisfies a power law (this holds for example if $K$ is the middle
third Cantor set). Thus it follows that the set of numbers in the middle third
Cantor set which are normal to no base has dimension $\log2/\log3$.