How many double squares can a string contain?

Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. In 1998 Fraenkel and Simpson showed that a string of length n contains at most 2n distinct squares. In 2007 Ilie provided an asymptotic upper bound of 2n−Θ(log n). We show that a string of length n contains at most b5n/3c 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 b2n/3c double squares.