TILING THE INTEGERS WITH TRANSLATES OF ONE FINITE SET

ETHAN M. COVEN AND AARON MEYEROWITZ

Abstract. A set tiles the integers if 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 [J. Number Theory 9 (1977), 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.