Avoiding abelian and additive powers in rich words

TL;DR

This paper constructs infinite rich words over small alphabets that avoid abelian and additive powers, advancing understanding of pattern avoidance in combinatorics on words.

Contribution

It introduces the first known infinite rich words over minimal alphabets that avoid additive powers of specific lengths.

Findings

Constructed an infinite additive 5-power-free rich word over {0,1}.

Constructed an infinite additive 4-power-free rich word over {0,1,2}.

Achieved minimal alphabet sizes for avoiding abelian and additive powers.

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.