Avoiding abelian powers cyclically

Abstract

We study a new notion of cyclic avoidance of abelian powers. A finite word w avoids abelian N-powers cyclically if for each abelian N-power of period m occurring in the infinite word w ω , we have m ≥ |w|. Let A(k) be the least integer N such that for all n there exists a word of length n over a k-letter alphabet that avoids abelian N-powers cyclically. Let A ∞ (k) be the least integer N such that there exist arbitrarily long words over a k-letter alphabet that avoid abelian N-powers cyclically.We prove that 5 ≤ A(2) ≤ 8, 3 ≤ A(3) ≤ 4, 2 ≤ A(4) ≤ 3, and A(k) = 2 for k ≥ 5. Moreover, we show that A ∞ (2) = 4, A ∞ (3) = 3, and A ∞ (4) = 2.