  Given a lattice $\Lambda$ in a locally compact abelian group $G$ and a
measurable subset $\Omega$ with finite and positive measure, then the set of
characters associated to the dual lattice form a frame for $L^2(\Omega)$ if and
only if the distinct translates by $\Lambda$ of $\Omega$ have almost empty
intersections. Some consequences of this results are the well-known Fuglede
theorem for lattices, as well as a simple characterization for frames of
modulates.
