Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. Fraenkel and Simpson showed in 1998 that a string of length n contains at most 2n distinct squares. Ilie presented in 2007 an asymptotic upper bound of 2n−Θ(log n). We show that a string of length n contains at most 11n/6 distinct squares. This new upper bound is obtained by investigating the combinatorial structure of double squares and showing that a string of length n contains at most 5n/6 particular double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpson's result.

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This article is about discrete periodicities and their combinatorial structures. It presents and describes the unique structure caused by the alteration of a pattern in a repetition. Those alterations of a pattern arise in the context of double squares and were discovered while working on bounding the number of distinct squares in a string. Nevertheless, they can arise in other phenomena and are worth being presented on their own.