Bounded lattice tiles that pack with another lattice

TL;DR

This paper constructs bounded sets that tile with one lattice and pack with another in Euclidean space, answering a question about lattice tilings and packings, and explores conditions under which such sets can be bounded or unbounded.

Contribution

It provides a method to construct bounded measurable sets that tile with one lattice and pack with another, addressing a question from 2001 and analyzing boundedness conditions.

Findings

Constructed bounded sets that tile with lattice L and pack with lattice M.

Showed such bounded sets cannot exist if lattices have different volumes.

Explored cases where unbounded sets can tile with both lattices.

Abstract

Suppose L and M are full-rank lattices in Euclidean space, such that vol(L) < 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. If we do not require measurability of F it is often possible that a set F can be found tiling with both L and M even when L and M have different volumes, for instance if their intersection is trivial. We also show here that such a set can never be bounded if L and M have different volumes.