Extremal overlap-free and extremal $\beta$-free binary words

TL;DR

This paper characterizes extremal overlap-free and $eta$-free binary words, identifying lengths that admit such words and demonstrating their existence for certain $eta$ values.

Contribution

It provides a complete characterization of lengths for extremal overlap-free binary words and proves the existence of arbitrarily long extremal $eta$-free words for specific $eta$ ranges.

Findings

Identifies all lengths with extremal overlap-free binary words.

Shows existence of arbitrarily long extremal $eta$-free words for $2^+ \\leq \\beta \\leq 8/3$.

Abstract

An overlap-free (or $\beta$-free) word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ at any position contains an overlap (or a factor of exponent at least $\beta$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every extended real number $\beta$ such that $2^+\leq\beta\leq 8/3$, we show that there are arbitrarily long extremal $\beta$-free binary words.