Abelian-Square-Rich Words

TL;DR

This paper investigates the abundance of abelian-square factors in infinite words, proving that certain well-known words like Thue-Morse and some Sturmian words are rich in these factors, with implications for combinatorics on words.

Contribution

It introduces the concepts of abelian-square-rich and uniformly abelian-square-rich words, and characterizes these properties for Thue-Morse and Sturmian words, including regularity results.

Findings

Thue-Morse word is uniformly abelian-square-rich.

Number of abelian-square factors in Thue-Morse is 2-regular.

Sturmian words with bounded partial quotients are uniformly abelian-square-rich.

Abstract

An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length $n$ grows quadratically with $n$. More precisely, we say that an infinite word $w$ is {\it abelian-square-rich} if, for every $n$, every factor of $w$ of length $n$ contains, on average, a number of distinct abelian-square factors that is quadratic in $n$; and {\it uniformly abelian-square-rich} if every factor of $w$ contains a number of distinct abelian-square factors that is proportional to the square of its length. Of course, if a word is uniformly abelian-square-rich, then it is abelian-square-rich, but we show that the converse is not true in general. We prove that the Thue-Morse word is uniformly abelian-square-rich and that the function counting the number of distinct abelian-square factors of length $2n$ of the Thue-Morse word is $2$-regular. As for Sturmian words, we prove that a Sturmian word $s_{\alpha}$ of angle $\alpha$ is uniformly abelian-square-rich if and only if the irrational $\alpha$ has bounded partial quotients, that is, if and only if $s_{\alpha}$ has bounded exponent.