On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne

Abstract

Let $F=\{\mathbf{p}_0,\ldots,\mathbf{p}_n\}$ be a collection of points in $\mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(\mathbf{x})=\lambda \mathbf{x} +(1-\lambda)\mathbf{p}_i,$$ where $\lambda \in(0,1)$. Given $F$ and $\lambda$ it is well known that there exists a unique non-empty compact set $X$ satisfying $X=\cup_{i=0}^n S_i(X)$. For each $\mathbf{x} \in X$ there exists a sequence $\mathbf{a}\in\{0,\ldots,n\}^{\mathbb{N}}$ satisfying $$\mathbf{x}=\lim_{j\to\infty}(S_{a_1}\circ \cdots \circ S_{a_j})(\mathbf{0}).$$ We call such a sequence a coding of $\mathbf{x}$. In this paper we prove that for any $F$ and $k \in\mathbb{N},$ there exists $\delta_k(F)>0$ such that if $\lambda\in(1-\delta_k(F),1),$ then every point in the interior of $X$ has a coding which is $k$-simply normal. Similarly, we prove that there exists $\delta_{uni}(F)>0$ such that if $\lambda\in(1-\delta_{uni}(F),1),$ then every point in the interior of $X$ has a coding containing all finite words. For some specific choices of $F$ we obtain lower bounds for $\delta_k(F)$ and $\delta_{uni}(F)$. We also prove some weaker statements that hold in the more general setting when the similarities in our iterated function systems exhibit different rates of contraction. Our proofs rely on a variation of a well known construction of a normal number due to Champernowne, and an approach introduced by Erd\H{o}s and Komornik.