Avoiding abelian and additive powers in rich words

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.