Optimal Bounds for the Similarity Density of the Thue-Morse Word with Overlap-Free and 73-Power-Free Infinite Binary Words

Abstract

We consider a measure of similarity for infinite words that generalizes the usual number-theoretic notion of asymptotic or natural density for subsets of natural numbers. We show that every [Formula: see text]-power-free infinite binary word, other than the Thue-Morse word t and its complement [Formula: see text], has this measure of similarity with t between [Formula: see text] and [Formula: see text], and that this bound is optimal in a strong sense just for the class of overlap-free words. This is a generalization of a classical 1927 result of Kurt Mahler.