Lattice tiling and the Weyl-Heisenberg frames

Abstract

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