Translational Tilings of the Integers with Long Periods
Suppose that A Z is a finite set of integers of diameter D=A - A. Suppose also that B Z is such that AB = Z , that is each nZ is uniquely expressible as a+b, aA, bB. 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) of B. We show that P(B) c_2 (c_3 D D D) and that we can have P(B) c_1 D^2, where c_1, c_2, c_3 > 0 are constants. , number=1, journal=The Electronic Journal of Combinatorics, author=Kolountzakis, Mihail N., year=2003, month=May, pages=#R22