Cube tilings with linear constraints

Abstract

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.