A Note on Squares in Binary Words

TL;DR

This paper investigates the structure of overlap-free binary words, establishing an upper bound on the number of middle positions of squares and demonstrating the existence of infinitely many words reaching this bound.

Contribution

It introduces a new bound on the middle positions of squares in overlap-free binary words and proves the existence of infinitely many words attaining this bound.

Findings

For overlap-free binary words, 2M(w) ≤ |w|+3.

Existence of infinitely many overlap-free words with 2M(w) = |w|+3.

Provides structural insights into squares within binary words.

Abstract

We consider words over a binary alphabet. A word $w$ is overlap-free if it does not have factors (blocks of consecutive letters) of the form $uvuvu$ for nonempty $u$. Let $M(w)$ denote the number of positions that are middle positions of squares in $w$. We show that for overlap-free binary words, $2M(w) \le |w|+3$, and that there are infinitely many overlap-free binary words for which $2M(w)=|w|+3$.