On a lemma of Crochemore and Rytter

Crochemore and Rytter introduced in 1995 a structural lemma on three squares starting at the same position. This influential lemma has been used by many researchers in the field of periodicities in strings. In particular, Fraenkel and Simpson used it in 1998 to obtain a universal upper bound for the maximum number of distinct squares occurring in a string. We present a generalization of Crochemore and Rytter's lemma by exploiting the combinatorics of two squares starting at the same position.