Another Way to Lower the Bound for Distinct Squares

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 $n-\mathrm{\Theta }(\log n)$ in [WORDS, 2023]. In this paper, we present another proof to obtain the same bound $n-\mathrm{\Theta }(\log n)$.