Another Way to Lower the Bound for Distinct Squares

Eitatsu Tomita; Tomohiro I

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arXiv · 2602.12711

Another Way to Lower the Bound for Distinct Squares

Eitatsu Tomita·Tomohiro I

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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 $n - \Theta(\log n)$.

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Eitatsu Tomita, Tomohiro I. 2026-02-13. Another Way to Lower the Bound for Distinct Squares. https://arxiv.org/abs/2602.12711

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