Tiling the integers with aperiodic tiles

Abstract

A finite subset A of integers tiles the discrete line ℤ if the integers can be written as a disjoint union of translates of A. In some cases, necessary and sufficient conditions for A to tile the integers are known. We extend this result to a large class of nonperiodic tilings and give a new formulation of the Coven–Meyerowitz reciprocity conjecture which is equivalent to Fuglede's conjecture in one dimension.