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

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

The growth rate and dimension theory of beta-expansions

In a recent paper of Feng and Sidorov they show that for $\beta\in(1,\frac{1+\sqrt{5}}{2})$ the set of $\beta$-expansions grows exponentially for every $x\in(0,\frac{1}{\beta-1})$. In this paper we study this growth rate further. We also consider the set of $\beta$-expansions from a dimension theory perspective

An analogue of Khintchine's theorem for self-conformal sets

Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFSs). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin-Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension. Combining our results with the mass transference principle of Beresnevich and Velani \cite{BerVel}, we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are "very well approximated". As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals. The ideas put forward in this paper are introduced in the general setting of IFSs that may contain overlaps. We believe that by viewing IFS's from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation