Cube tilings with linear constraints

Abstract

We consider tilings $(\mathcal{Q},Φ)$ of $\mathbb{R}^d$ where $\mathcal{Q}$ is the $d$-dimensional unit cube and the set of translations $Φ$ 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 $Φ$ is a sublattice of $A\mathbb{Z}^d$ with any $d \in \mathbb{N}$ or a generic subset of $A\mathbb{Z}^d$ with $d\leq 7$. As a direct consequence of our results, we obtain a criterion for the existence of linearly constrained frequency sets, that is, $Φ\subseteq A\mathbb{Z}^d$, such that the respective set of complex exponential functions $\mathcal{E} (Φ)$ 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.

Figures (1)

• Figure 1: We display the projection of the cubes $[0,1)^4$, $[0,1)^4+\gamma$, $[0,1)^4+\xi$, $[0,1)^4+\eta$, $[0,1)^4+\alpha$, $[0,1)^4+\beta$, $[0,1)^4+\tau$ along $e_4$. This scenario in the case of $d=3$ is achieved, for example, by starting with the canonical tiling $([0,1)^4,\mathbb{Z}^4)$ and then shifting disjoint columns -- one in $x$ direction, one in $y$ direction, and one in $z$ direction -- that neighbor the origin in $x$ neighboring by a non integer.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2: Minkowski Mi1907, Hajós Ha42, Keller Keller1930, Perron Pe40aPe40b, Brakensiek et al. BHMN20
• Theorem 3
• Remark 4
• Theorem 6: Kolountzakis
• Corollary 7
• Theorem 8
• Lemma 9
• Remark 10
• proof : Proof of Lemma \ref{['lem:cubetiling']}