Lattice tiling and the Weyl-Heisenberg frames (Q5955169)

It is proved that for two arbitrary full-rank lattices \(L,K\) in \(R^d\) having fundamental sets with equal volume, there exists a set \(\Omega\) which tiles \(R^d\) by both \(L\) and \(K\). The similar result for more than two lattices is proved to be false. As a consequence of the results in the paper, a density result for Gabor frames in \(L^2(R^d)\) is proved. It is a generalization of a well-known one-dimensional result (namely that a necessary condition for \(\{e^{2\pi i mbx}g(x-na)\}_{m,n\in Z}\) to be a frame for \(L^2(R)\) is that \(ab\leq 1\)).