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.