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We consider tilings (Q, ) of Rᵈ where Q is the d-dimensional unit cube and the set of translations is constrained to lie in a pre-determined lattice A Zᵈ in Rᵈ. We provide a full characterization of matrices A for which such cube tilings exist when is a sublattice of AZᵈ with any d N or a generic subset of AZᵈ with d 7. As a direct consequence of our results, we obtain a criterion for the existence of linearly constrained frequency sets, that is, AZᵈ, such that the respective set of complex exponential functions E () is an orthogonal Fourier basis for the space of square integrable functions supported on a parallelepiped BQ, where A, B R^d d are nonsingular matrices given a priori. Similarly constructed Riesz bases are considered in a companion paper.
Lee et al. (Tue,) studied this question.