Matolcsi Mate: Weak tiling by polytopes in $\mathbb R^d$

Abstract: 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 $\mathbb R^d$ such that $1_A \ast \mu =1_{A^c}$. 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