Intersecting periodic words

Let a = a[1..2v] be a word which is a square and for which a[1..2u] is also a square and a[k + 1..min(k + 2w, 2v)] has period w where u < w < v < 2u and 0 ≤ k ≤ v − u. We show that under certain extra conditions on k and w the word has period gcd(u, v, w) and describe the word’s structure when these extra conditions do not hold. c © 2006 Elsevier B.V. All rights reserved.