Avoidability of long $k$-abelian repetitions

TL;DR

This paper investigates the avoidability of long $k$-abelian repetitions in infinite words over binary and ternary alphabets, providing new results on what patterns can or cannot be avoided.

Contribution

It establishes new avoidability thresholds for $k$-abelian-squares and cubes, and answers open questions related to abelian repetitions in infinite words.

Findings

Cannot avoid abelian-cubes of period at least 2 in infinite binary words.

Can avoid 3-abelian-squares of period at least 3 in binary words.

Can avoid 2-abelian-squares of period at least 2 in ternary words.

Abstract

We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are M\"akel\"a's questions. We show that one cannot avoid abelian-cubes of abelian period at least $2$ in infinite binary words, and therefore answering negatively one question from M\"akel\"a. Then we show that one can avoid $3$-abelian-squares of period at least $3$ in infinite binary words and $2$-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct $k$-abelian-squares that must appear in an infinite binary word.