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Suppose that A Z is a finite set of integers of diameter D =m axA minA. Suppose also that B Z is such that A B = Z ,t hat is eachn 2 Z is uniquely expressible as a + b, a 2 A, b 2 B. We say then that A tiles the integers if translated at the locations B and it is well known that B must be a periodic set in this case and that the smallest period of B is at most 2 D . Here we study the relationship between the diameter of A and the least period P(B )o fB .W e show that P(B) c2 exp(c3 p D logD p log log D) and that we can have P(B) c1D 2 , where c1,c2,c3 > 0 are constants.

Translational Tilings of the Integers with Long Periods