# How Many Square Occurrences Must a Binary Sequence Contain?

### Abstract

Every binary word with at least four letters contains a square. A. Fraenkel and J. Simpson showed that three *distinct squares* are necessary and sufficient to construct an infinite binary word. We study the following complementary question: how many *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 $0.55080...$.