The lexicographically least square-free word with a given prefix

TL;DR

This paper investigates the structure of the lexicographically least square-free infinite words with a given prefix, revealing connections to the ruler sequence and providing morphisms for their generation.

Contribution

It introduces a detailed analysis of $L(p)$, the least square-free word with prefix $p$, and provides morphisms to generate these words for various prefixes.

Findings

$L(p)$ reflects the structure of the ruler sequence for certain prefixes.

Morphisms are provided to generate $L(n)$ for specific letters and two-letter prefixes.

The structure of $L(p)$ is more complex for non-empty prefixes, with partial characterizations given.

Abstract

The lexicographically least square-free infinite word on the alphabet of non-negative integers with a given prefix $p$ is denoted $L(p)$. When $p$ is the empty word, this word was shown by Guay-Paquet and Shallit to be the ruler sequence. For other prefixes, the structure is significantly more complicated. In this paper, we show that $L(p)$ reflects the structure of the ruler sequence for several words $p$. We provide morphisms that generate $L(n)$ for letters $n=1$ and $n\geq3$, and $L(p)$ for most families of two-letter words $p$.