Lattice tiling and the Weyl—Heisenberg frames

Abstract.

Let {\cal L} and {\cal K} be two full rank lattices in ${\Bbb R}^d$. We prove that if ${\rm v}({\cal L} ) = {\rm v}({\cal K})$, i.e. they have the same volume, then there exists a measurable set $\Omega$ such that it tiles ${\Bbb R}^d$ by both ${\cal L}$ and ${\cal K}$. A counterexample shows that the above tiling result is false for three or more lattices. Furthermore, we prove that if ${\rm v}({\cal L}) \le {\rm v}({\cal K})$ then there exists a measurable set $\Omega$ such that it tiles by ${\cal L}$ and packs by ${\cal K}$. Using these tiling results we answer a well-known question on the density property of Weyl—Heisenberg frames.