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

Abstract

Given b > 1 and y ∈ R/Z, we consider the set of x ∈ R such that y is not a limit point of the sequence {bnx mod 1 : n ∈ 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 ⊂ R (that is, supporting measures μ satisfying certain decay conditions). Furthermore, the intersection has full dimension in K if μ 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 log 2/ log 3.