A note on the number of squares in a word

Fraenkel and Simpson [A.S. Fraenkel, J. Simpson, How many squares can a string contain? J. Combin. Theory Ser. A 82 (1998) 112–120] proved that the number of squares in a word of length n is bounded by 2n. In this note we improve this bound to 2n−Θ(logn). Based on the numerical evidence, the conjectured bound is n.