Abstract
We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes 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.