Cube tilings with linear constraints

We consider tilings (Q,Φ) of Rᵈ where Q is the d-dimensional unit cube and the set of translations Φ 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 Φ 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, Φ ⊆ AZᵈ, such that the respective set of complex exponential functions E (Φ) is an orthogonal Fourier basis for the space of square integrable functions supported on a parallelepiped BQ, where A, B ∈ Rd × d are nonsingular matrices given a priori. Similarly constructed Riesz bases are considered in a companion paper.