On the number of squares in a finite word

TL;DR

This paper proves a bound on the number of distinct square factors in a finite word, confirming a conjecture by Fraenkel and Simpson from 1998.

Contribution

It establishes a tight upper bound on the number of square factors in finite words based on their length and alphabet size, resolving a longstanding conjecture.

Findings

Bound of |w|-|Alphabet(w)|+1 on square factors

Confirmed the Fraenkel and Simpson conjecture

Provides a new combinatorial insight into word structure

Abstract

A {\em square} is a word of the form $uu$. In this paper we prove that for a given finite word $w$, the number of distinct square factors of $w$ is bounded by $|w|-|\Alphabet(w)|+1$, where $|w|$ denotes the length of $w$ and $|\Alphabet(w)|$ denotes the number of distinct letters in $w$. This result answers a conjecture of Fraenkel and Simpson stated in 1998.