Partial words and the interaction property of periods

A theorem of Fine and Wilf expresses the interaction property of periods, which is a basic property of periodic words. An arbitrary word with given periods $p$ and $q$ also has a “derived” period $\operatorname{gcd}(p,q)$ if the length of the word is greater than some critical value called the length of interaction. In this paper we consider a similar property for arbitrary periodic partial words and give a sharp linear bound for the length of interaction.