Similarity density of the Thue-Morse word with overlap-free infinite binary words

Abstract

We consider a measure of similarity for infinite words that generalizes the notion of asymptotic or natural density of subsets of natural numbers from number theory. We show that every overlap-free infinite binary word, other than the Thue-Morse word t and its complement t, has this measure of similarity with t between 1 4 and 3 4 . This is a partial generalization of a classical 1927 result of Mahler.