Periodicity and local complexity of Delone sets

2025-04-29

AI Summary

This paper investigates the relationship between periodicity and local complexity of Delone sets in ℝᵈ, modeled as indicator-function-valued ℝᵈ-configurations. Method: Introducing, for the first time in the continuous setting, the notions of annihilators and periodizers of configurations, the authors employ multidimensional symbolic dynamics, Laurent polynomial algebra, and geometric analysis. Results: They prove that low ℝᵈ-pattern complexity implies the existence of nontrivial annihilators—hence algebraic constraints; establish that slow patch complexity characterizes annihilability for Meyer sets; and derive the first finitary local-complexity criterion enforcing periodicity for colored Delone sets. The core contribution is the extension of algebraic dynamical systems methods to continuous spaces, thereby uncovering a fundamental triadic linkage among complexity, algebraic constraints, and periodicity—providing a novel paradigm for assessing order in aperiodic structures.

Technology Category

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• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

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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 $c: mathbb{Z}^d o mathcal{A}$ 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 $c: mathbb{R}^d o mathcal{A}$ 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 $c_1,ldots,c_m: mathbb{R}^d o mathbb{Z}$. 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.

Problem

• Study periodicity and complexity of Delone sets using algebraic methods
• Generalize annihilators and periodizers for multidimensional configurations
• Analyze pattern complexity and forced periodicity in colored Delone sets

Innovation

• Algebraic approach for Delone sets analysis
• Generalized annihilators for R^d-configurations
• Low complexity implies annihilators existence