Bounded lattice tiles that pack with another lattice

Abstract

Suppose $L,M$ are full-rank lattices in Euclidean space, such that $\mathrm{vol}L<\mathrm{vol}M$. Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set $F$ (we can even take $F$ to be a finite union of polytopes) such that $F+L$ is a tiling and $F+M$ is a packing. In the related problem of finding a common fundamental domain for two lattices $L$ and $M$ we show that no such common fundamental domain exists if the volumes are different even, if we do not require measurability of this domain.

14 pages, 2 figures; the proof of the intermediate case has now been corrected in v3 to use induction on the dimension. Theorem 2.1 added in v3