The Coven–Meyerowitz tiling conditions for 3 odd prime factors

Abstract

It is well known that if a finite set A⊂Z tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A⊕B=ZM of a finite cyclic group. We are interested in characterizing all finite sets A⊂Z that have this property. Coven and Meyerowitz (J Algebra 212:161–174, 1999) proposed conditions (T1), (T2) that are sufficient for A to tile, and necessary when the cardinality of A has at most two distinct prime factors. They also proved that (T1) holds for all finite tiles, regardless of size.