Tilings of the integers can have superpolynomial periods

Abstract

Let A be a finite set of integers. We say that A tiles the integers if there is a set T ⊆ ℤ such that {t+A: tT{ forms a disjoint partition of the integers. It has long been known that such a set T must be periodic. The question is to determine how long the period of T can become as a function of the diameter of the set A. The previous best lower bound, due to Kolountzakis [7], shows that the period of T can grow as fast as the square of the diameter of A. In this paper we improve Kolountzakis’ lower bound by showing that the period of T can in fact grow faster than any power of the diameter of A.

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Steinberger, J.P. Tilings of the integers can have superpolynomial periods. Combinatorica 29, 503–509 (2009). https://doi.org/10.1007/s00493-009-2269-9

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Mathematics Subject Classification (2000)

  • 20K25