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 of length $n$ that uses $σ$ distinct letters is at most $n - σ$. In this paper, we give a new upper bound $n - Θ(\log n)$, improving the previous bound when $σ\in o(\log n)$.

翻译：平方串是指形如$xx$的字符串，其中$x$为非空字符串。Brlek与Li[Comb. Theory, 2025]证明了长度为$n$且包含$σ$个不同字母的字符串中，不同平方串的数量至多为$n - σ$。本文提出一个新的上界$n - Θ(\log n)$，该结果在$σ\in o(\log n)$时改进了先前的上界。