Square-free extensions of words

TL;DR

This paper investigates the properties and generation of square-free words over finite alphabets, exploring extremal cases and recursive extension methods, with computational evidence supporting the existence of infinite square-free words over alphabets of size three or more.

Contribution

It introduces new insights into the structure of extremal square-free words and proposes a recursive extension method, supported by computational experiments, to generate infinite square-free words.

Findings

Existence of infinitely many extremal square-free words over a 3-letter alphabet.

Conjecture that no extremal words exist over a 4-letter alphabet.

Computer experiments suggest recursive extension produces infinite square-free words over alphabets of size at least three.

Abstract

A word is square-free if it does not contain nonempty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over a $3$-letter alphabet. It was proved recently [7] that among these words there are infinitely many extremal ones, that is, having a square in every single-letter extension. We study diverse problems concerning extensions of words preserving the property of avoiding squares. Our main motivation is the conjecture stating that there are no extremal words over a $4$-letter alphabet. We also investigate a natural recursive procedure of generating square-free words by a single-letter right-most extension. We present the results of computer experiments supporting a supposition that this procedure gives an infinite square-free word over any alphabet of size at least three.