Another Way to Lower the Bound for Distinct Squares

TL;DR

This paper offers an alternative proof for the upper bound on the number of distinct squares in a word, showing it is at most n minus a logarithmic term, refining understanding of word structure.

Contribution

It provides a new proof for the existing bound on distinct squares, simplifying or offering a different perspective on the previous results.

Findings

The number of distinct squares in a word is at most n - Θ(log n).

The new proof confirms the same bound as previous work.

Provides a different approach to bounding distinct squares.

Abstract

A square is a word of the form $xx$ for a non-empty word $x$. Brlek and Li [Comb. Theory, 2025] proved that the number of distinct squares in a word $w$ of length $n$ is at most $n - \sigma$, where $\sigma$ is the number of letters used in $w$. The same authors extended the proof to lower the upper bound to $n - \Theta(\log n)$ in [WORDS, 2023]. In this paper, we present another proof to obtain the same bound $n - \Theta(\log n)$.