Avoidability of long k-abelian repetitions

We study the avoidability of long k-abelian-squares and k-abeliancubes on binary and ternary alphabets. For k = 1, these are Mäkelä’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 Mäkelä. 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.