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.