Abstract : Every binary word with at least four letters contains a square. A.~Fraenkel and J.~Simpson \cite{FraenkelSimpson95} showed that three {\em distinct squares} are necessary and sufficient to construct an infinite binary word. We study the following complementary question: how many {\em square occurrences} must a binary word contain? We show that this quantity is, in the limit, a constant fraction of the word length, and prove that this constant is $0.55080...$.
Gregory Kucherov, Pascal Ochem, Michael Rao. How many square occurrences must a binary sequence contain?. The Electronic Journal of Combinatorics, Open Journal Systems, 2003, 10 (1), 11 p. ⟨inria-00099596⟩