Avoiding Fractional Powers over the Natural Numbers

Abstract

We study the lexicographically least infinite $a/b$-power-free word on the alphabet of non-negative integers. Frequently this word is a fixed point of a uniform morphism, or closely related to one. For example, the lexicographically least $7/4$-power-free word is a fixed point of a $50847$-uniform morphism. We identify the structure of the lexicographically least $a/b$-power-free word for three infinite families of rationals $a/b$ as well many "sporadic" rationals that do not seem to belong to general families. To accomplish this, we develop an automated procedure for proving $a/b$-power-freeness for morphisms of a certain form, both for explicit and symbolic rational numbers $a/b$. Finally, we establish a connection to words on a finite alphabet. Namely, the lexicographically least $27/23$-power-free word is in fact a word on the finite alphabet $\{0, 1, 2\}$, and its sequence of letters is $353$-automatic.