An extension of the periodicity lemma to longer periods Original Research Article

The well-known Periodicity Lemma of Fine and Wilf states that if the word x of length n has periods image satisfying image, then x has also period d, where image. Here we study the structure of long periods, namely image. In particular, we construct recursively a sequence of integers image, and show that such x is covered with periods image by a basic kernel factor. We further compute the maximum alphabet size image of A over which a word with long periods can exist, and compute the subword complexity of x over this A.