Avoidability of long 𝑘-abelian repetitions

Abstract

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