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

TL;DR

This paper uses Schmidt games to show that certain sets of numbers, which are not well-approximated by base-b sequences, intersect with fractals like the Cantor set in a way that preserves their full Hausdorff dimension.

Contribution

It establishes that these non-normality sets intersect fractals with full dimension using Schmidt game techniques and measures satisfying decay conditions.

Findings

Sets of numbers not normal to any base intersect fractals with full dimension.

The intersection with the middle third Cantor set has dimension rac{rac{2}{rac{3}}}.

Schmidt game methods extend to bi-Lipschitz images of these sets.

Abstract

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 $\log2/\log3$.