Periodic structure of translational multi-tilings in the plane

["Suppose f epsilon L-1(R-d), A subset of R-d is a finite union of translated lattices such that f + A tiles with a weight. We prove that there exists a lattice L subset of R-d such that f + L also tiles, with a possibly different weight. As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity.","Our second result is a new characterization of convex polygons that multi-tile the plane by translations. It also provides a very efficient criteria to determine whether a convex polygon admits translational multi-tilings. As an application, one can easily construct symmetric (2m)-gons, for any m >= 4, that do not multi-tile by translations.","Finally, we prove a convex polygon which is not a parallelogram only admits periodic multiple tilings, if any."]