A k-abelian cube is a word uvw, where the factors u, v, and w are either pairwise equal, or have the same multiplicities for every one of their factors of length at most k. Previously it has been shown that k-abelian cubes are avoidable over a binary alphabet for k ≥ 8. Here it is proved that this holds for k ≥ 5.

We give analogs of the complexity $p\left(n\right)$ and of Sturmian words which are called respectively the $*$-complexity $p_{*}\left(n\right)$ and $*$-Sturmian words. We show that the class of $*$-Sturmian words coincides with the class of words satisfying $p_{*}\left(n\right)\le n+1$, and we determine the structure of $*$-Sturmian words. For a class of words satisfying $p_{*}\left(n\right)=n+1$, we give a general formula and an upper bound for $p\left(n\right)$. Using this general formula, we give explicit formulae for $p\left(n\right)$ for some words belonging to this class. In general, $p\left(n\right)$ can take large values, namely,...