1 4 A pr 1 99 9 On the structure of multi PREPRINT ple translational tilings by polygonal regions

We consider polygons with the following “pairing property”: for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon K tiles the plane multiply when translated at the locations Λ, where Λ is a multiset in the plane. The pairing property of K makes this question particularly amenable to Fourier Analysis. After establishing a necessary and sufficient condition for K to tile with a given lattice Λ (which was first found by Bolle for the case of convex polygons–notice that all convex polygons that tile, necessarily have the pairing property and, therefore, our theorems apply to them) we move on to prove that a large class of such polygons tiles only quasi-periodically, which for us means that Λ must be a finite union of translated 2-dimensional lattices in the plane. For the particular case of convex polygons we show that all convex polygons which are not parallelograms tile necessarily quasi-periodically, if at all. §0. Introduction In this paper we study multiple tilings of the plane by translates of a polygonal region of a certain type, the polygons with the pairing property of Definition 2 below. Definition 1 (Tiling) Let K be a measurable subset of R2 of finite measure and let Λ ∈ R2 be a discrete multiset (i.e., its underlying set is discrete and each point has finite multiplicity). We say that K +Λ is a (translational, multiple) tiling of R2, if ∑ λ∈Λ 1K(x− λ) = w, for almost all (Lebesgue) x ∈ R2, where the weight or level w is a positive integer and 1K is the indicator function of K. Definition 2 (Polygons with the Pairing Property) A polygon K has the pairing property if for each edge e there is precisely one other edge of K parallel to e Remarks. 1. Note that all symmetric convex polygons have the pairing property and it is not hard to see that all convex polygons that tile by translation are necessarily symmetric. 2. The polygonal regions we deal with are not assumed to be connected. Partially supportedby the U.S. National Science Foundation, under grant DMS 97-05775.