Tiling the Integers with Translates of One Finite Set

Abstract

A settiles the integersif and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers. For sets of prime power size, it was solved by D. Newman (1977,J. Number Theory9, 107–111). We solve it for sets of size having at most two prime factors. The conditions are always sufficient, but it is unknown whether they are necessary for all finite sets.