On the number of $k$-powers in a finite word

TL;DR

This paper provides an upper bound on the number of k-powers in finite words by analyzing special factors, addressing a longstanding conjecture related to the structure of repeated patterns.

Contribution

It introduces a novel counting method using special factors to bound the number of k-powers in finite words, advancing understanding of word combinatorics.

Findings

Upper bound established for k-powers in finite words

Method based on counting right-special factors

Addresses a conjecture by Fraenkel and Simpson

Abstract

This note is an attempt to attack a conjecture of Fraenkel and Simpson stated in 1998 concerning the number of distinct squares in a finite word. By counting the number of (right-)special factors, we give an upper bound of the number of {\em $k$-powers} in a finite word for any integer $k\geq 3$. By {\em $k$-power}, we mean a word of the form $\underbrace{uu...u}_{k \; \text{times}}$.