An Extension of the Periodicity Lemma to Longer Periods (Invited Lecture)

Abstract

The well-known periodicity lemma of Fine and Wilf states that if the word x of length n has periods p, q satisfying p + q - d ≤ n, then x has also period d, where d = gcd(p, q). Here we study the case of long periods, namely p+q -d > n, for which we construct recursively a sequence of integers p = p1 > p2 >...> pj-1 ≥ 2, such that x1, up to a certain prefix of x1, has these numbers as periods. We further compute the maximum alphabet size ¦A¦ = p+q -n of A over which a word with long periods can exist, and compute the subword complexity of x over A.