Translational Tilings of the Integers with Long Periods

Suppose that A ⊆ Z is a £nite set of integers of diameter D = max A−min A. Suppose also that B ⊆ Z is such that A ⊕ B = Z, that is each n ∈ Z is uniquely expressible as a + b, a ∈ A, b ∈ 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 2D. Here we study the relationship between the diameter of A and the least period P(B) of B. We show that P(B) ≤ c2 exp(c3 √ D log D √ log log D) and that we can have P(B) ≥ c1D, where c1, c2, c3 > 0 are constants. Subject Classi£cation: Primary 11B75, Secondary 10A25. Notation. Let G be an abelian group denoted additively. If A, B ⊆ G we write A + B = {a + b : a ∈ A, b ∈ B}. We also write A + b in place of A + {b} to denote a translate of set A. If in the set A + B every element is written uniquely as a + b, with a ∈ A and b ∈ B we may write A⊕ B in place of A + B. If A, B, E ⊆ G and E = A⊕ B we say that A tiles E when translated by B (and similarly that B tiles E when translated by A). We also say that A⊕ B is a tiling of E by A (or B). Sets A for which there is B ⊆ G such that G = A⊕ B are called tiles. For every positive integer n we write [n] = {0, 1, . . . , n− 1}. Subsets of [n] will also be viewed as subsets of Zn = Z/(nZ), the additive group of residues modn. For any integer n and a set X ⊆ G we write nX = {nx : x ∈ X}. ∗Supported in part by European Commission IHP Network HARP (Harmonic Analysis and Related Problems), Contract Number: HPRN-CT-2001-00273 HARP.