On the structure of multi

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 , where is a multiset in the plane. The pairing property of K makes this question particularly amenable to Fourier Analysis. After establishing a necessary and suucient condition for K to tile with a given lattice (which was rst 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 must be a nite 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. x0. 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 Deenition 2 below. Deenition 1 (Tiling) Let K be a measurable subset of R 2 of nite measure and let 2 R 2 be a discrete multiset (i.e., its underlying set is discrete and each point has nite multiplicity). We say that K + is a (translational, multiple) tiling of R 2 , if X 2 1 K (x ?) = w; for almost all (Lebesgue) x 2 R 2 , where the weight or level w is a positive integer and 1 K is the indicator function of K. 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.