Delone Sets Research Articles

On the Regularity Radius of Delone Sets in $${\mathbb {R}}^3$$

We complete the proof of the upper bound $\hat\rho_3\leq 10R$ for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all $10R$-clusters of a Delone set $X$ with parameters $(r,R)$ are equivalent, then $X$ is a regular system.

Chaotic Delone sets

<p style='text-indent:20px;'>We present a definition of chaotic Delone set and establish the genericity of chaos in the space of <inline-formula><tex-math id="M1">\begin{document}$ (\epsilon,\delta) $\end{document}</tex-math></inline-formula>-Delone sets for <inline-formula><tex-math id="M2">\begin{document}$ \epsilon\geq \delta $\end{document}</tex-math></inline-formula>. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets.

Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.

Delone Sets Generated by Square Roots

Delone sets are locally finite point sets, such that (a) any two points are separated by a given minimum distance, and (b) there is a given radius so that every ball of that radius contains at least one point. Important examples include the vertex set of Penrose tilings and other regular model sets, which serve as a mathematical model for quasicrystals. In this note, we show that the point set given by the values with is a Delone set in the complex plane, for any . This complements Akiyama’s recent observation (see Spiral Delone sets and three distance theorem (2020), Nonlinearilty, 33(5): 2533–2540) that with forms a Delone set, if and only if α is badly approximated by rationals. A key difference is that our setting does not require Diophantine conditions on α.

Bounded displacement non-equivalence in substitution tilings

In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially to results concerning instances of non-equivalence. We present a general condition for two Delone sets to be BD non-equivalent, and apply our result to Delone sets associated with tilings of Euclidean space. First we consider substitution tilings, and exhibit a substitution matrix associated with two distinct substitution rules. The first rule generates only periodic tilings, while the second generates tilings for which any associated Delone set is non-equivalent to any lattice in space. As an extension of this result, we introduce arbitrarily many distinct substitution rules associated with a single matrix, with the property that Delone sets generated by distinct rules are non-equivalent. We then turn to the study of mixed substitution tilings, and present a mixed substitution system that generates representatives of continuously many distinct BD equivalence classes.

Delone sets in ℝ3: Regularity Conditions

A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.

Spiral Delone sets and three distance theorem

We show that a constant angle progression on the Fermat spiral forms a Delone set if and only if its angle is badly approximable.

Equilibrium Configurations for Generalized Frenkel–Kontorova Models on Quasicrystals

I study classes of generalized Frenkel–Kontorova models whose potentials are given by almost-periodic functions which are closely related to aperiodic Delone sets of finite local complexity. Since such Delone sets serve as models for quasicrystals, this setup presents Frenkel–Kontorova models for the type of aperiodic crystals which have been discovered since Shechtman’s discovery of quasicrystals. Here I consider models with configurations $$u:{\mathbb {Z}}^r \rightarrow {\mathbb {R}}^d$$ , where d is the dimension of the quasicrystal, for any r and d. The almost-periodic functions used for potentials are called pattern-equivariant and I show that if the interactions of the model satisfies a mild $$C^2$$ requirement, and if the potential satisfies a mild non-degeneracy assumption, then there exist equilibrium configurations of any prescribed rotation rotation number/vector/plane. The assumptions are general enough to satisfy the classical Frenkel–Kontorova models and its multidimensional analoges. The proof uses the idea of the anti-integrable limit.

A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation

We define a general framework that includes objects such as tilings, Delone sets, functions, and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object mathcal {P} to one that is MLD with mathcal {P}. Moreover, we show that, for an object mathcal {P} and a class Sigma of objects, a mild condition on them ensures that there exists some mathcal {Q}in Sigma that is MLD with mathcal {P}. As an application, we study pattern-equivariant functions. In particular, we show that the space of all pattern-equivariant functions contains all the information on the original object up to MLD, in a quite general setting.

Index Theory and Topological Phases of Aperiodic Lattices

We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.

On substitution tilings and Delone sets without finite local complexity

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [ 29 ] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.

Delone dynamical systems and spectral convergence

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

On the origin of crystallinity: a lower bound for the regularity radius of Delone sets.

The mathematical conditions for the origin of long-range order or crystallinity in ideal crystals are one of the very fundamental problems of modern crystallography. It is widely believed that the (global) regularity of crystals is a consequence of `local order', in particular the repetition of local fragments, but the exact mathematical theory of this phenomenon is poorly known. In particular, most mathematical models for quasicrystals, for example Penrose tiling, have repetitive local fragments, but are not (globally) regular. The universal abstract models of any atomic arrangements are Delone sets, which are uniformly distributed discrete point sets in Euclidean d space. An ideal crystal is a regular or multi-regular system, that is, a Delone set, which is the orbit of a single point or finitely many points under a crystallographic group of isometries. The local theory of regular or multi-regular systems aims at finding sufficient local conditions for a Delone set X to be a regular or multi-regular system. One of the main goals is to estimate the regularity radius \hat{\rho}_d for Delone sets X in terms of the radius R of the largest `empty ball' for X. The celebrated `local criterion for regular systems' provides an upper bound for \hat{\rho_d} for any d. Better upper bounds are known for d ≤ 3. The present article establishes the lower bound \hat{\rho_d}\geq 2dR for all d, which is linear in d. The best previously known lower bound had been \hat{\rho}_d\geq 4R for d ≥ 2. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent (2dR - ℇ)-clusters, which are not regular systems. The two- and three-dimensional constructions are illustrated by examples. In addition to its fundamental importance, the obtained result is also relevant for the understanding of geometrical conditions of the formation of ordered and disordered arrangements in polytypic materials.

Delone Sets in ℝ3 with 2R-Regularity Conditions

A regular system is the orbit of a point with respect to a crystallographic group. The central problem of the local theory of regular systems is to determine the value of the regularity radius, which is the least number such that every Delone set of type (r,R) with identical neighborhoods/clusters of this radius is regular. In this paper, conditions are described under which the regularity of a Delone set in three-dimensional Euclidean space follows from the pairwise congruence of small clusters of radius 2R. Combined with the analysis of one particular case, this result also implies the proof of the “10R-theorem,” which states that if the clusters of radius 10R in a Delone set are congruent, then this set is regular.

Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula

We study operators defined on a Hilbert space defined by a self-affine Delone set $$\Lambda $$ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain $$\limsup $$ law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or $${\mathbb {R}}^d$$ -invariant distributions of a dynamical system defined by $$\Lambda $$ . We use this to refine Shubin’s trace formula for certain self-adjoint operators acting on $$\ell ^2(\Lambda )$$ and show that the errors of convergence in Shubin’s formula are given by these traces.

The Local Theory for Regular Systems in the Context of t-Bonded Sets

The main goal of the local theory for crystals developed in the last quarter of the 20th Century by a geometry group of Delone (Delaunay) at the Steklov Mathematical Institute is to find and prove the correct statements rigorously explaining why the crystalline structure follows from the pair-wise identity of local arrangements around each atom. Originally, the local theory for regular and multiregular systems was developed with the assumption that all point sets under consideration are ( r , R ) -systems or, in other words, Delone sets of type ( r , R ) in d-dimensional Euclidean space. In this paper, we will review the recent results of the local theory for a wider class of point sets compared with the Delone sets. We call them t-bonded sets. This theory, in particular, might provide new insight into the case for which the atomic structure of matter is a Delone set of a “microporous” character, i.e., a set that contains relatively large cavities free from points of the set.

Non-rectifiable Delone sets in SOL and other solvable groups

National Science Foundation DMS-1207296 CONICYT's Anillo Research Project 1103 DySyRF CNRS UMR 8628 ERC 257110 RaWG

ON SELF-SIMILARITIES OF CUT-AND-PROJECT SETS

Among the commonly used mathematical models of quasicrystals are Delone sets constructed using a cut-and-project scheme, the so-called cut-and-project sets. A cut-and-project scheme (&lt;em&gt;L&lt;/em&gt;,π&lt;sub&gt;1&lt;/sub&gt;, π&lt;sub&gt;2&lt;/sub&gt;) is given by a lattice &lt;em&gt;L&lt;/em&gt; in R&lt;sup&gt;s&lt;/sup&gt; and projections π&lt;sub&gt;1&lt;/sub&gt;, π&lt;sub&gt;2&lt;/sub&gt; to suitable subspaces V&lt;sub&gt;1&lt;/sub&gt;, V&lt;sub&gt;2&lt;/sub&gt;. In this paper we derive several statements describing the connection between self-similarity transformations of the lattice &lt;em&gt;L&lt;/em&gt; and transformations of its projections π&lt;sub&gt;1&lt;/sub&gt;(&lt;em&gt;L&lt;/em&gt;), π&lt;sub&gt;2&lt;/sub&gt;(&lt;em&gt;L&lt;/em&gt;). For a self-similarity of a set Σ we take any linear mapping A such that AΣ ⊂ Σ, which generalizes the notion of self-similarity usually restricted to scaled rotations. We describe a method of construction of cut-and-project scheme such that π&lt;sub&gt;1&lt;/sub&gt;(&lt;em&gt;L&lt;/em&gt;) ⊂ R&lt;sup&gt;2&lt;/sup&gt; is invariant under an isometry of order 5. We describe all linear self-similarities of the scheme thus constructed and show that they form an 8-dimensional associative algebra over the ring Z. We perform an example of a cut-and-project set with linear self-similarity which is not a scaled rotation.

Self Affine Delone Sets and Deviation Phenomena

We study the growth of norms of ergodic integrals for the translation action on spaces coming from expansive, self-affine Delone sets. The linear map giving the self-affinity induces a renormalization map on the pattern space and we show that the rate of growth of ergodic integrals is controlled by the induced action of the renormalizing map on the cohomology of the pattern space up to boundary errors. We explore the consequences for the diffraction of such Delone sets, and explore in detail what the picture is for substitution tilings as well as for cut and project sets which are self-affine. We also explicitly compute some examples.

Calibrated Configurations for Frenkel–Kontorova Type Models in Almost Periodic Environments

The Frenkel–Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We focus, in this context, on the existence of calibrated configurations (a notion stronger than the standard minimizing condition). In any dimension and for any continuous superlinear interaction energies, we exhibit a set, called projected Mather set, formed of environments that admit calibrated configurations. In the one-dimensional setting, we then give sufficient conditions on the family of interaction energies that guarantee the existence of calibrated configurations for every environment. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry–Mather theory and spaces of Delone sets.