[Paper Review] Linearly repetitive Delone sets are rectifiable

José Aliste‐Prieto, Daniel Coronel+1 more|arXiv (Cornell University)|Mar 28, 2011

Quasicrystal Structures and Properties12 references4 citations

TL;DR

This paper proves that every linearly repetitive Delone set in $\mathbb{R}^d$ for $d \geq 2$ is bi-Lipschitz equivalent to the integer lattice $\mathbb{Z}^d$, establishing a strong geometric rigidity for these aperiodic structures. For primitive substitution tilings, it identifies a spectral condition on the substitution matrix—specifically, eigenvalues satisfying a Pisot-type criterion—that ensures bounded displacement equivalence to a scaled lattice $\beta\mathbb{Z}^d$, extending beyond classical Pisot systems.

ABSTRACT

In this paper we prove that, for any integer $d>0$, every linearly repetitive Delone set in the Euclidean $d$-space $\RR^d$ is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice $\ZZ^d$. In the particular case when the Delone set $X$ in $\RR^d$ comes from a primitive substitution tiling of $\RR^d$, we give a condition on the eigenvalues of the substitution matrix which implies the existence of a homeomorphism with bounded displacement from $X$ to the lattice lattice $λ\ZZ^d$ for some positive $λ$. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.

Motivation & Objective

• To resolve the long-standing question of whether all linearly repetitive Delone sets in $\mathbb{R}^d$ ($d \geq 2$) are bi-Lipschitz equivalent to $\mathbb{Z}^d$.
• To extend known results on bounded displacement equivalence from primitive Pisot substitution tilings to a broader class of substitution systems.
• To establish a geometric link between the dynamical properties of Delone sets and their metric equivalence to lattices.
• To provide a unified framework for understanding rectifiability of aperiodic Delone sets via combinatorial and spectral conditions on substitution rules.

Proposed method

• The authors define linearly repetitive Delone sets via uniform control on the frequency of patches within balls, using the notion of repetition radius.
• They employ a hierarchical decomposition of Delone sets based on substitution rules, analyzing the growth of boundary patches across scales.
• A key technical tool is the use of a matrix $M$ encoding substitution rules, with spectral properties (eigenvalues) governing the scaling behavior of patches.
• The proof relies on estimating the discrepancy between the number of patches in a region and its expected volume via a constant $K$ depending on the Delone set's parameters.
• They apply a recursive estimate on the number of boundary patches $\mathcal{L}(\mathcal{T}^{l+1}, \partial U)$ across scales, bounded by $\mathcal{L}(\mathcal{T}, \partial U)$ and geometric scaling factors.
• The final bound is derived using a geometric summation argument over scales, combining estimates from small and large scales to yield a uniform discrepancy control.

Experimental results

Research questions

• RQ1 Is every linearly repetitive Delone set in $\mathbb{R}^d$ ($d \geq 2$) bi-Lipschitz equivalent to $\mathbb{Z}^d$?
• RQ2 What spectral condition on the substitution matrix ensures bounded displacement equivalence between a Delone set and a scaled lattice $\beta\mathbb{Z}^d$?
• RQ3 Can the class of substitution tilings admitting bounded displacement to $\mathbb{Z}^d$ be extended beyond the classical Pisot type?
• RQ4 How does the repetition radius of a Delone set constrain its metric equivalence to a lattice?
• RQ5 To what extent do spectral properties of the substitution matrix control the regularity of the associated Delone set?

Key findings

• Every linearly repetitive Delone set in $\mathbb{R}^d$ ($d \geq 2$) is bi-Lipschitz equivalent to $\mathbb{Z}^d$, resolving a key question in geometric rigidity of aperiodic sets.
• For primitive substitution tilings in $\mathbb{R}^d$, bounded displacement equivalence to $\beta\mathbb{Z}^d$ holds if the substitution matrix has eigenvalues satisfying a Pisot-type condition.
• The bound on discrepancy between patch counts and expected volumes is controlled by a constant $K$ depending only on the Delone set, independent of the region size.
• The proof establishes a uniform estimate $|\mathcal{N}(\mathcal{T}, U) - \alpha \mu_d(U)| \leq K \mathcal{L}(\mathcal{T}, \partial U)$, where $\mathcal{L}$ counts boundary patches.
• The method extends beyond Pisot systems, showing that the class of substitution tilings with bounded displacement to a lattice is strictly larger than the Pisot class.
• The result confirms that linear repetitivity implies strong metric regularity, implying that such Delone sets are geometrically indistinguishable from lattices up to bi-Lipschitz distortion.