Lyndon Words, the Three Squares Lemma, and Primitive Squares

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 $n \log_ϕn \approx 1.441n\log_2 n$, where $ϕ$ is the golden ratio, reported by Fraenkel and Simpson [TCS 1999] obtained via the Three Squares Lemma.

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