Avoiding abelian and additive powers in rich words

TL;DR

This work investigates the avoidability of abelian and additive powers within rich words. It constructs two infinite fixed points of affine morphisms, $B=eta^wedge(0)$ over $0,1$ and $oldsymbol{ extGamma}=gamma^wedge(1)$ over $0,1,2$, and proves they are respectively additive $5$-power-free and additive $4$-power-free. Additive power-freeness is established via the Currie–Mol–Rampersad–Shallit decision algorithm for fixed points of affine morphisms, while richness is shown for $B$using Walnut (owing to $5$-uniformity) and for $oldsymbol{ extGamma}$ via a palindrome-preservation lemma and a unioccurrent palindromic suffix construction. The results give the smallest possible alphabets achieving these additive-power-free rich words and raise compelling open problems about the existence of infinite additive power-free rich words over other integer subsets, as well as about the corresponding abelian cases and maximum finite lengths in related settings.

Abstract

This paper concerns the avoidability of abelian and additive powers in infinite rich words. In particular, we construct an infinite additive $5$-power-free rich word over $\{0,1\}$ and an infinite additive $4$-power-free rich word over $\{0, 1, 2\}$. The alphabet sizes are as small as possible in both cases, even for abelian powers.

Figures (1)

• Figure 1: The prefix $P$ of $\mathbf{\Gamma}$

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4: Currie, Mol, Rampersad, and Shallit CurrieMolRampersadShallit2021
• Proposition 3.1
• proof
• Proposition 3.2
• proof