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

TL;DR

This paper introduces a measure of similarity for infinite binary words and demonstrates that all overlap-free words, except the Thue-Morse word and its complement, have a similarity measure with Thue-Morse between 1/4 and 3/4, extending Mahler's classical result.

Contribution

It generalizes the concept of similarity measure to infinite words and characterizes the similarity range of overlap-free words with Thue-Morse, except for two specific cases.

Findings

Overlap-free words other than Thue-Morse have similarity between 1/4 and 3/4.

The measure extends classical number theory density concepts to infinite words.

Thue-Morse and its complement are unique in their similarity measure with themselves.

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 bar, 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.