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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 iteratedfunction 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 uniquenon-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$$\mathbf{x}=\lim_{j\to\infty}(S_{a_1}\circ \cdots \circS_{a_j})(\mathbf{0}).$$ We call such a sequence a coding of $\mathbf{x}$. Inthis 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 inthe interior of $X$ has a coding which is $k$-simply normal. Similarly, weprove that there exists $\delta_{uni}(F)>0$ such that if$\lambda\in(1-\delta_{uni}(F),1),$ then every point in the interior of $X$ hasa coding containing all finite words. For some specific choices of $F$ weobtain lower bounds for $\delta_k(F)$ and $\delta_{uni}(F)$. We also prove someweaker statements that hold in the more general setting when the similaritiesin our iterated function systems exhibit different rates of contraction. Ourproofs rely on a variation of a well known construction of a normal number dueto Champernowne, and an approach introduced by Erd\H{o}s and Komornik.

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