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. © 1999 Academic Press.