Digit frequencies and self-affine sets with non-empty interior

Abstract

In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $\beta\in(1,1.787\ldots)$ then every $x\in(0,\frac{1}{\beta-1})$ has a simply normal $\beta$-expansion. We also prove that if $\beta\in(1,\frac{1+\sqrt{5}}{2})$ then every $x\in(0,\frac{1}{\beta-1})$ has a $\beta$-expansion for which the digit frequency does not exist, and a $\beta$-expansion with limiting frequency of zeros $p$, where $p$ is any real number sufficiently close to $1/2$. For a class of planar self-affine sets we show that if the horizontal contraction lies in a certain parameter space and the vertical contractions are sufficiently close to $1,$ then every nontrivial vertical fibre contains an interval. Our approach lends itself to explicit calculation and give rise to new examples of self-affine sets with non-empty interior. One particular strength of our approach is that it allows for different rates of contraction in the vertical direction.