Problems in between words and abelian words: k-abelian avoidability

We consider a recently defined notion of k-abelian equivalence of words in connection with avoidability problems. This equivalence relation, for a fixed natural number k, takes into account the numbers of occurrences of the different factors of length k and the prefix and the suffix of length k−1. We search for the smallest alphabet in which k-abelian squares and cubes can be avoided, respectively. For 2-abelian squares this is four–as in the case of abelian words, while for 2-abelian cubes we have only strong evidence that the size is two–as it is in the case of words. However, we are able to prove this optimal value only for 8-abelian cubes.