Schmidt’s game, fractals, and numbers normal to no base

Given $b > 1$ and $y \in \R/\Z$, we consider the set of $x\in \R$ 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 \hd, 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 $K\subset \R$ (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 $\log2/\log3$.

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Published 1 January 2010