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Lattice sub-tilings and frames in LCA groups

Given a lattice $Λ$ in a locally compact abelian group $G$ and a measurable subset $Ω$ with finite and positive measure, then the set of characters associated to the dual lattice form a frame for $L^2(Ω)$ if and only if the distinct translates by $Λ$ of $Ω$ 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.