Avoiding abelian and additive powers in rich words

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.