Lyndon Words, the Three Squares Lemma, and Primitive Squares

TL;DR

This paper generalizes the Three Squares Lemma using Lyndon words and improves the upper bound on the number of primitive squares in a string to n log_2 n, enhancing understanding of string combinatorics.

Contribution

It introduces a more general variant of the Three Squares Lemma and provides a tighter upper bound on primitive squares in strings, both based on Lyndon words.

Findings

Generalized the Three Squares Lemma with Lyndon words

Established an upper bound of n log_2 n for primitive squares

Improved upon the previous bound of approximately 1.441n log_2 n

Abstract

We revisit the so-called "Three Squares Lemma" by Crochemore and Rytter [Algorithmica 1995] and, using arguments based on Lyndon words, derive a more general variant which considers three overlapping squares which do not necessarily share a common prefix. We also give an improved upper bound of $n\log_2 n$ on the maximum number of (occurrences of) primitively rooted squares in a string of length $n$, also using arguments based on Lyndon words. To the best of our knowledge, the only known upper bound was $n \log_\phi n \approx 1.441n\log_2 n$, where $\phi$ is the golden ratio, reported by Fraenkel and Simpson [TCS 1999] obtained via the Three Squares Lemma.