Cube Tilings with Linear Constraints

We consider tilings \((\mathcal {Q},\Phi )\) $(\mathcal{Q},\mathrm{\Phi })$ of \(\mathbb {R}^d\) ${\mathbb{R}}^d$ where \(\mathcal {Q}\) $\mathcal{Q}$ is the d-dimensional unit cube and the set of translations \(\Phi \) $\mathrm{\Phi }$ is constrained to lie in a pre-determined lattice \(A \mathbb {Z}^d\) $A{\mathbb{Z}}^d$ in \(\mathbb {R}^d\) ${\mathbb{R}}^d$ . We provide a full characterization of matrices A for which such cube tilings exist when \(\Phi \) $\mathrm{\Phi }$ is a sublattice of \(A\mathbb {Z}^d\) $A{\mathbb{Z}}^d$ with any \(d \in \mathbb {N}\) $d\in \mathbb{N}$ or a generic subset of \(A\mathbb {Z}^d\) $A{\mathbb{Z}}^d$ with \(d\le 7\) $d\le 7$ . As a direct consequence of our results, we obtain a criterion for the existence of linearly constrained frequency sets, that is, \(\Phi \subseteq A\mathbb {Z}^d\) $\mathrm{\Phi }\subseteq A{\mathbb{Z}}^d$ , such that the respective set of complex exponential functions \(\mathcal {E} (\Phi )\) $\mathcal{E}(\mathrm{\Phi })$ is an orthogonal Fourier basis for the space of square integrable functions supported on a parallelepiped \(B\mathcal {Q}\) $B\mathcal{Q}$ , where \(A, B \in \mathbb {R}^{d \times d}\) $A,B\in {\mathbb{R}}^{d\times d}$ 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).