Abelian Anti-Powers in Infinite Words

An abelian anti-power of order $k$ (or simply an abelian $k$-anti-power) is a concatenation of $k$ consecutive words of the same length having pairwise distinct Parikh vectors. This definition generalizes to the abelian setting the notion of a $k$-anti-power, as introduced in [G. Fici et al., Anti-powers in infinite words, J. Comb. Theory, Ser. A, 2018], that is a concatenation of $k$ pairwise distinct words of the same length. We aim to study whether a word contains abelian $k$-anti-powers for arbitrarily large $k$. S. Holub proved that all paperfolding words contain abelian powers of every order [Abelian powers in paper-folding words. J. Comb. Theory, Ser. A, 2013]. We show that they also contain abelian anti-powers of every order.