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. Fraenkel and Simpson?showed in 1998 that a string of length n contains at most 2 n distinct squares. Ilie presented in 2007 an asymptotic upper bound of 2 n - ? ( log n ) . We show that a string of length n contains at most ? 11 n / 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 ? 5 n / 6 ? particular double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpson's result.