Avoiding fractional powers over the natural numbers

TL;DR

This paper investigates the structure of the lexicographically least infinite words over non-negative integers that avoid fractional powers, revealing fixed points of morphisms and connections to automatic sequences.

Contribution

It identifies the structure of these words for various rational ratios and develops an automated method to prove power-freeness of morphic words.

Findings

The least $7/4$-power-free word is a fixed point of a 50847-uniform morphism.

The structure of words for three families of rationals is characterized.

The $27/23$-power-free word over a finite alphabet is 353-automatic.

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.