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arXiv · 2609.32084

The Tight Upper Bound on the Number of Distinct Squares in Circular Words

Rikuya Hamai
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Abstract

A square is a word $xx$, where $x$ is nonempty. We show that a circular word of length $n$ contains at most $\lfloor 3n/2 \rfloor$ distinct squares of length at most $n$. The proof combines known results relating squares to circuits in Rauzy graphs. The coefficient $3/2$ agrees with the known lower bound.

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Rikuya Hamai. 2026-09-25. The Tight Upper Bound on the Number of Distinct Squares in Circular Words. https://arxiv.org/abs/2609.32084

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