Translational tilings of the integers with long periods
Combinatorics
2007-05-23 v1 Number Theory
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.
Cite
@article{arxiv.math/0210476,
title = {Translational tilings of the integers with long periods},
author = {Mihail N. Kolountzakis},
journal= {arXiv preprint arXiv:math/0210476},
year = {2007}
}
Comments
6 pages, 2 figures