Multi-tiling and equidecomposability of polytopes by lattice translates

Abstract

We characterize the polytopes in ${\mathbb{R}}^d$ (not necessarily convex or connected ones) which multi-tile the space by translations along a given lattice. We also give a necessary and sufficient condition for two polytopes in ${\mathbb{R}}^d$ to be equidecomposable by lattice translations.

To appear in the Bulletin of the London Mathematical Society