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Translational tilings of the integers with long periods

Mihail N. Kolountzakis
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Abstract

Suppose that A is a finite set of integers of diameter D. Suppose also that the set of integers B is such that A+B is a tiling of the integers, that is each integer is uniquely expressible as a+b, with a in A, b in B. It is well known that B must be a periodic set in this case. Here we study the relationship between the diameter D of A and the least period T of B. We show that T is at most C exp(C \sqrt D \log D \sqrt{\log\log D}) and that we can have T at least quadratic in D.

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Mihail N. Kolountzakis. 2002-10-31. Translational tilings of the integers with long periods. https://arxiv.org/abs/math/0210476

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