Periodicity and local complexity of Delone sets

TL;DR

The paper develops an algebraic framework for studying periodicity and local complexity of Delone sets by extending multidimensional symbolic dynamics from $\Z^d$ to $\R^d$-configurations. It shows that low $\R^d$-pattern complexity or slow patch growth forces nontrivial annihilators, and that in several cases these annihilators yield explicit decompositions into finitely many periodic components; in particular, Meyer sets with slow complexity possess annihilators, and 1D Delone configurations with annihilators are necessarily periodic. The authors also establish a periodic decomposition theorem and a forced periodicity criterion under finite local complexity, connecting local structure to global regularity. Collectively, the results provide a robust algebraic mechanism to infer periodicity from local data in higher-dimensional aperiodic order, with implications for understanding quasicrystals and related tilings.

Abstract

We study complexity and periodicity of Delone sets by applying an algebraic approach to multidimensional symbolic dynamics. In this algebraic approach, $\mathbb{Z}^d$-configurations $c: \mathbb{Z}^d \to \mathcal{A}$ for a finite set $\mathcal{A} \subseteq \mathbb{C}$ and finite $\mathbb{Z}^d$-patterns are regarded as formal power series and Laurent polynomials, respectively. In this paper we study also functions $c: \mathbb{R}^d \to \mathcal{A}$ where $\mathcal{A}$ is as above. These functions are called $\mathbb{R}^d$-configurations. Any Delone set may be regarded as an $\mathbb{R}^d$-configuration by simply presenting it as its indicator function. Conversely, any $\mathbb{R}^d$-configuration whose support (that is, the set of cells for which the configuration gets non-zero values) is a Delone set can be seen as a colored Delone set. We generalize the concept of annihilators and periodizers of $\mathbb{Z}^d$-configurations for $\mathbb{R}^d$-configurations. We show that if an $\mathbb{R}^d$-configuration has a non-trivial annihilator, that is, if a linear combination of some finitely many of its translations is the zero function, then it has an annihilator of a particular form. Moreover, we show that $\mathbb{R}^d$-configurations with integer coefficients that have non-trivial annihilators are sums of finitely many periodic functions $c_1,\ldots,c_m: \mathbb{R}^d \to \mathbb{Z}$. Also, $\mathbb{R}^d$-pattern complexity is studied alongside with the classical patch-complexity of Delone sets. We point out the fact that sufficiently low $\mathbb{R}^d$-pattern complexity of an $\mathbb{R}^d$-configuration implies the existence of non-trivial annihilators. Moreover, it is shown that if a Meyer set has sufficiently slow patch-complexity growth, then it has a non-trivial annihilator. Finally, a condition for forced periodicity of colored Delone sets of finite local complexity is provided.

Figures (3)

• Figure 1: The Delone sets defined in Example \ref{['ex: strict hierarchy']}.
• Figure 2: The open half space $H_{\mathbf{v}}$ in direction $\mathbf{v} = (-1,3)$. The black line corresponds to the set $\overline{H}_{\mathbf{v}} \setminus H_{\mathbf{v}} = \langle \mathbf{v} \rangle ^{\perp}$.
• Figure 3: Illustration of the proof of Theorem \ref{['thm: forced periodicity Delone FLC']}. The smaller ball is the ball $B_{T_0}(-T_0 \mathbf{v})$ and the bigger ball is the ball $B_{T}( -T \mathbf{v}'_{n})$ for some $T \geq T_0$ and $n \geq N_0$.

Theorems & Definitions (83)

• Definition 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Lemma 3.3: icalp
• Theorem 3.4: icalp, fullproofs
• Theorem 3.5: Periodic decomposition theorem fullproofs
• Lemma 3.6: The dilation lemma fullproofs
• Lemma 3.7: Adapted from fullproofs
• proof