Title: On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne
Abstract: Let F={p0,…,pn} be a collection of points in Rd. The set F naturally gives rise to a family of iterated function systems consisting of contractions of the formSi(x)=λx+(1−λ)pi,i=0,…,n, where λ∈(0,1) and x∈Rd. Given F and λ it is well known that there exists a unique non-empty compact set X satisfying X=∪i=0nSi(X). For each x∈X there exists a sequence (aj)j=1∞∈{0,…,n}N satisfyingx=limj→∞(Sa1∘⋯∘Saj)(0). We call such a sequence a coding of x. In this paper we prove that for any F and k∈N, there exists δk(F)>0 such that if λ∈(1−δk(F),1), then every point in the interior of X has a coding which is k-simply normal. Similarly, we prove that there exists δuni(F)>0 such that if λ∈(1−δuni(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 δk(F) and δuni(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 Erdős and Komornik.