Cube Tilings with Linear Constraints

We consider tilings \((\mathcal {Q},\Phi )\) of \(\mathbb {R}^d\) where \(\mathcal {Q}\) is the d-dimensional unit cube and the set of translations \(\Phi \) is constrained to lie in a pre-determined lattice \(A \mathbb {Z}^d\) in \(\mathbb {R}^d\). We provide a full characterization of matrices A for which such cube tilings exist when \(\Phi \) is a sublattice of \(A\mathbb {Z}^d\) with any \(d \in \mathbb {N}\) or a generic subset of \(A\mathbb {Z}^d\) with \(d\le 7\). As a direct consequence of our results, we obtain a criterion for the existence of linearly constrained frequency sets, that is, \(\Phi \subseteq A\mathbb {Z}^d\), such that the respective set of complex exponential functions \(\mathcal {E} (\Phi )\) is an orthogonal Fourier basis for the space of square integrable functions supported on a parallelepiped \(B\mathcal {Q}\), where \(A, B \in \mathbb {R}^{d \times d}\) are nonsingular matrices given a priori. Similarly constructed Riesz bases are considered in a companion paper (Lee et al., Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice, 2024. arXiv:2401.08042).