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

Details Schmidt's game, fractals, and numbers normal to no base Schmidt's game, fractals, and numbers normal to no base

2009-09-23

arxiv.org/abs/0909.4251v3

Math. Res. Letters 17 (2010), 307-321

Mathematics

Dynamical Systems

15 pages; minor corrections made

Scientific paper

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 $\log2/\log3$.

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