An upper bound of the number of distinct powers in binary words

TL;DR

This paper proves a conjecture that the number of distinct powers, including squares, in binary words is bounded by a specific function of the word's length and letter counts, refining previous bounds.

Contribution

It establishes a stronger upper bound on the number of distinct powers in binary words, confirming a conjecture by Jonoska, Manea, and Seki.

Findings

Proves the conjectured upper bound for binary words.

Provides a tighter bound than previously known.

Enhances understanding of repetitive structures in binary words.

Abstract

A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer and a square is a word of the form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a word is bounded by the length of the word. This conjecture was proven recently by Brlek and Li. Besides, there exists a stronger upper bound for binary words conjectured by Jonoska, Manea and Seki stating that for a word of length $n$ over the alphabet $\left\{a, b\right\}$, if we let $k$ be the least of the number of a's and the number of b's and $k \geq 2$, then the number of distinct squares is upper bounded by $\frac{2k-1}{2k+2}n$. In this article, we prove this conjecture by giving a stronger statement on the number of distinct powers in a binary word.