Titre : How many double squares can a string contain?
Exposant : Frantisek Franek
Résumé : Counting the number of types of squares rather than their occurrences, we consider the problem of bounding the maximum 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\-\Theta(\log\ n). We show that a string of length n contains at most \lfloor11n/6\rfloor 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 \lfloor5n/6\rfloor double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpson's result. Joint work with Antoine Deza and Adrien Thierry.