A Tighter Upper Bound for the Number of Distinct Squares in Circular Words

TL;DR

This paper improves the upper bound on the number of distinct square factors in circular words from 1.8n to 5/3 n, advancing towards the conjectured bound of 1.5n.

Contribution

It provides a tighter upper bound of 5/3 n for the number of distinct squares in circular words, refining previous bounds.

Findings

Previous bound was 1.8n, now improved to 5/3 n.

Supports the conjecture that the bound is 1.5n.

Advances understanding of square factors in circular words.

Abstract

A \emph{square} is a word of the form $uu$, where $u$ is a nonempty finite word. Given a finite word $w$ of length $n$, let $[w]$ denote the corresponding \emph{circular word}, i.e., the set of all cyclic rotations of $w$. We study the number of distinct square factors of the elements of $[w]$. Amit and Gawrychowski first showed that this number is upper bounded by $3.14n$. In a recent article, Charalampopoulos et al. improved this upper bound to $1.8n$ and conjectured that the sharp upper bound is $1.5n$. In this note, we improve this upper bound to $\frac{5}{3}n$.