On a generalization of Abelian equivalence and complexity of infinite words

Abstract

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 N , 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 k : N → N and show that if P (k) ω (n) < q k (n) for some k ∈ Z + ∪ {+∞} and n ≥ 0, the ω is ultimately periodic. Moreover if ω is aperiodic, then P (k) ω (n) = q k (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.

Another Generalization of Abelian Equivalence: Binomial Complexity of Infinite Words

“…In general, for k-abelian equivalence, the number of equivalence classes for words of length n over a ℓ-letter alphabet is Θ(n (ℓ−1)ℓ k−1 ) [8]. We consider similar results for m-binomial equivalence (proofs can be found in [15]).…”

“…As usual, |u| denotes the length of u and |u| x denotes the number of occurrences of the word x as a factor of the word u. Karhumäki et al [8] introduce the notion of k-abelian equivalence of finite words as follows. Let u, v be two words over A.…”

Variations of the Morse-Hedlund Theorem for k-Abelian Equivalence

“…Abelian equivalence of words has long been a subject of great interest (see, for instance, Erdős's problem, [5,6,7,9,17,22,23,24,26]). Although the notion of k-abelian equivalence is quite new, there are already a number of papers on the topic [11,12,13,14,15,18].…”

“…As it turns out, each intermediate complexity function P k w can be used to detect periodicity of words. As a starting point of our research, we list two classical results on factor and abelian complexity in connection with periodicity, and their k-abelian counterparts proved by the authors in [15]. We note that in each case, the first two items are included in the third.…”

“…For instance, in the context of one-sided infinite words, the first item in Theorem 1 gives rise to a characterization of ultimately periodic words, while for the other two, the result holds in only one direction: If P k w (n) < min{n+1, 2k} for some k ∈ Z + and n ≥ 1 then w is ultimately periodic, but not conversely (see [15]). For instance in the simplest case when k = 1, it is easy to see that if w is the ultimately periodic word 01 ω , then for each positive integer n there are precisely two abelian classes of factors of w of length n. However, the same is true of the (aperiodic) infinite Fibonacci word w = 0100101001001 · · · defined as the fixed point of the morphism 0 → 01, 1 → 0.…”