Weak tiling by polytopes in R^d

We say that a set A of positive measure weakly tiles its complement A^c, if there exists a positive, locally finite Borel measure \mu on R^d such that 1_A * \mu =1_{A^c}. We prove that if a (not necessarily convex) polytope A weakly tiles its complement by translations, then A is equidecomposable by translations to a cube of the same volume.

Joint work with M. Kolountzakis and N. Lev