Distinct Squares in Circular Words

TL;DR

This paper investigates the maximum number of distinct squares in all cyclic rotations of a circular word of length n, establishing new upper and lower bounds that extend classical results from linear words.

Contribution

It introduces the problem of counting distinct squares in circular words and provides the first non-trivial bounds, including an upper bound of 3.14n and a lower bound of 1.25n.

Findings

Upper bound of 3.14n for the number of distinct squares in circular words.

Existence of an infinite family of words with at least 1.25n distinct squares.

Extension of the classical linear word problem to circular words.

Abstract

A circular word, or a necklace, is an equivalence class under conjugation of a word. A fundamental question concerning regularities in standard words is bounding the number of distinct squares in a word of length $n$. The famous conjecture attributed to Fraenkel and Simpson is that there are at most $n$ such distinct squares, yet the best known upper bound is $1.84n$ by Deza et al. [Discr. Appl. Math. 180, 52-69 (2015)]. We consider a natural generalization of this question to circular words: how many distinct squares can there be in all cyclic rotations of a word of length $n$? We prove an upper bound of $3.14n$. This is complemented with an infinite family of words implying a lower bound of $1.25n$.