The Lexicographically Least Binary Rich Word Achieving the Repetition Threshold

Abstract

A word is rich if each of its length $n$ factors contains $n$ distinct non-empty palindromes. For a language ${\mathcal L}$, the repetition threshold of ${\mathcal L}$ is defined by
$$\text{RT}({\mathcal L})=\sup\{k: \text{ every infinite word of ${\mathcal L}$ contains a $k$-power}\}.$$
Currie et al. (2020) proved that the repetition threshold for binary rich words is $2+\sqrt{2}/2$. We exhibit the lexicographically least infinite binary rich word attaining this threshold.