Another Way to Lower the Bound for Distinct Squares

Eitatsu Tomita; Tomohiro I

Another Way to Lower the Bound for Distinct Squares

Discrete Mathematics 2026-03-03 v2

Authors: Eitatsu Tomita , Tomohiro I

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

Cite

@article{arxiv.2602.12711,
  title  = {Another Way to Lower the Bound for Distinct Squares},
  author = {Eitatsu Tomita and Tomohiro I},
  journal= {arXiv preprint arXiv:2602.12711},
  year   = {2026}
}

Comments

The manuscript has been revised to include an important reference that was missing in the previous version

Another Way to Lower the Bound for Distinct Squares

Abstract

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