  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 $\{b^n x \bmod
1: n\in\mathbb{N}\}$. 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
$\log2/\log3$.
