Avoiding abelian powers cyclically
Jarkko Peltomäki, Markus A. Whiteland · Advances in Applied Mathematics · 2020
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 wω, 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.