Geodesic
Partial words and the interaction property of periods
Izvestiya. Mathematics , Tome 68 (2004) no. 2, pp. 405-428

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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. 
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A. M. Shur; Yu. V. Gamzova. Partial words and the interaction property of periods. Izvestiya. Mathematics , Tome 68 (2004) no. 2, pp. 405-428. http://geodesic.mathdoc.fr/item/IM2_2004_68_2_a8/