On a generalization of Abelian equivalence and complexity of infinite words

In this paper we introduce and study a family of complexity functions of infinite words indexed by k ∈ Z+ ∪ {+∞}. Let k ∈ Z+ ∪ {+∞} and A be a finite non-empty set. Two finite words u and v in A are said to be k-Abelian equivalent if for all x ∈ A of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations ∼k on A , bridging the gap between the usual notion of Abelian equivalence (when k = 1) and equality (when k = +∞). We show that the number of k-Abelian equivalence classes of words of length n grows polynomially, although the degree is exponential in k. Given an infinite word ω ∈ A, we consider the associated complexity function P (k) ω : N → N which counts the number of k-Abelian equivalence classes of factors of ω of length n. We show that the complexity function P(k) is intimately linked with periodicity. More precisely we define an auxiliary function q : N → N and show that if P (k) ω (n) < q(n) for some k ∈ Z+ ∪ {+∞} and n ≥ 0, the ω is ultimately periodic. Moreover if ω is aperiodic, then P (k) ω (n) = q(n) if and only if ω is Sturmian. We also study k-Abelian complexity in connection with repetitions in words. Using Szemerédi’s theorem, we show that if ω has bounded k-Abelian complexity, then for every D ⊂ N with positive upper density and for every positive integer N, there exists a k-Abelian N power occurring in ω at some position j ∈ D.