Translational tilings of the integers with long periods (Q1408525)

Summary: Suppose that \(A \subseteq \mathbb Z\) is a finite set of integers of diameter \(D= \text{max} A - \min A\). Suppose also that \(B \subseteq \mathbb Z\) is such that \( A\oplus B = \mathbb Z\), that is each \(n \in \mathbb Z\) is uniquely expressible as \(a+b, a \in A, b \in B\). 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 \(P(B)\) of \(B\). We show that \(P(B) \leq c_2 \exp (c_3 \sqrt D \log D \sqrt {\log\log D})\) and that we can have \(P(B) \geq c_1 D^2\), where \( c_1, c_2, c_3 > 0\) are constants.