Avoiding 5/4-Powers on the Alphabet of Nonnegative Integers

Abstract

We identify the structure of the lexicographically least word avoiding $5/4$-powers on the alphabet of nonnegative integers. Specifically, we show that this word has the form $\mathbf{p} \, \tau ( \varphi(\mathbf{z}) \varphi^2(\mathbf{z}) \cdots)$ where $\mathbf{p},\mathbf{z}$ are finite words, $\varphi$ is a $6$-uniform morphism, and $\tau$ is a coding. This description yields a recurrence for the $i$th letter, which we use to prove that the sequence of letters is $6$-regular with rank $188$. More generally, we prove $k$-regularity for a sequence satisfying a recurrence of the same type.