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Delone set

A Delone set, also spelled Delaunay set, is a discrete subset \Lambda of Euclidean space \mathbb{R}^d that is both uniformly discrete and relatively dense, characterized by positive constants r and R (with r \leq R) such that the distance between any two distinct points in \Lambda is at least r, and every open ball of radius R centered at any point in \mathbb{R}^d intersects \Lambda nontrivially.^[1] This structure ensures that the points are neither too close together nor leave arbitrarily large gaps in space, providing a balanced model for evenly distributed point configurations.^[2] Named after the Russian mathematician Boris Delone (also known as Delaunay), the concept was introduced in the late 1930s by the Russian school of crystallographers as a foundational tool for modeling atomic arrangements in solid-state materials.^[3] In this context, Delone sets abstract the ideal positions of atoms in a crystal lattice, where uniform discreteness captures minimal interatomic distances due to repulsion, and relative density prevents voids larger than a certain scale.^[4] The original work emphasized their role in geometric crystallography, linking local point patterns to global symmetry properties.^[1] Beyond classical crystals, Delone sets have proven essential in studying aperiodic order, particularly in quasicrystals—materials discovered in the 1980s that exhibit diffraction patterns with sharp peaks but lack translational periodicity.^[5] Subclasses like Meyer sets (Delone sets whose difference set \Lambda - \Lambda is also Delone) and finite-type Delone sets model these structures by imposing finite local complexity, enabling the analysis of long-range correlations without periodicity.^[6] Applications extend to dynamical systems, where Delone sets underpin hull constructions for ergodic theory and diffraction measures; to tiling theory, via dual relationships with Voronoi diagrams and Delaunay triangulations; and to computational geometry, including bounds on regularity radii and chromatic mosaics.^[3]^[7] These properties make Delone sets a versatile framework for discrete geometry and mathematical physics.^[8]

Definitions and Properties

Formal Definition

A Delone set, also sometimes referred to as a Delaunay set, is a subset \Lambda of the Euclidean space \mathbb{R}^d that is both uniformly discrete and relatively dense. There exist positive constants r \leq R < \infty such that (i) every pair of distinct points in \Lambda is separated by a distance of at least $2r, and (ii) every point in \mathbb{R}^d lies within distance R of at least one point in \Lambda.^[9]^[4]^[10] Uniform discreteness formalizes the notion that points in \Lambda cannot cluster too closely and is expressed as

\inf \{ \|x - y\| : x, y \in \Lambda, \, x \neq y \} \geq 2r > 0.

The factor of $2r in the minimal separation distance is conventional, aligning with the packing radius r, which ensures that no open ball of radius r centered anywhere in \mathbb{R}^d contains more than one point of \Lambda.^[4]^[10] Relative density ensures that \Lambda covers \mathbb{R}^d without large gaps and is equivalently stated as

\sup \{ \|x - y\| : x \in \mathbb{R}^d, \, y \in \Lambda \} \leq R < \infty,

meaning every ball of radius R in \mathbb{R}^d contains at least one point of \Lambda. This condition assumes basic familiarity with the Euclidean metric \|\cdot\| and open or closed balls in \mathbb{R}^d.^[9]^[4] Although the parameters r and R—representing the largest uniform discreteness constant and smallest relative density constant, respectively—may vary in notation across texts, this choice is standard in the mathematical literature on point sets.^[10]

Key Properties

Delone sets are named after the Russian mathematician Boris Delone (1890–1980), who introduced the concept in the 1930s within mathematical crystallography to model atomic arrangements; they are also referred to as (r, R)-systems.^[11] A fundamental property of a Delone set \Lambda \subset \mathbb{R}^d is that its difference set \Lambda - \Lambda = \{ x - y : x, y \in \Lambda \} is itself a Delone set, hence locally finite in the sense of being both discrete and relatively dense in \mathbb{R}^d.^[3] The uniform discreteness of \Lambda with parameter r > 0 ensures that \Lambda - \Lambda has no accumulation points other than possibly at 0, and since |x - y| \geq 2r for distinct x, y \in \Lambda, the origin is isolated, making \Lambda - \Lambda discrete with minimum distance $2r. The relative density of \Lambda with parameter R > 0 implies that \Lambda - \Lambda is relatively dense with covering radius $2R, as for any z \in \mathbb{R}^d, the ball B(0, R) contains some y \in \Lambda and B(z, R) contains some x \in \Lambda, so x - y \in B(z, 2R).^[12] Delone sets possess positive lower density and finite upper density, providing uniform bounds on their asymptotic point counts. The upper density satisfies \overline{\delta}(\Lambda) \leq \frac{1}{\kappa_d r^d}, arising from the non-overlapping balls of radius r/2 centered at points of \Lambda, where \kappa_d denotes the volume of the unit ball in \mathbb{R}^d. The lower density satisfies \underline{\delta}(\Lambda) \geq \frac{1}{\kappa_d R^d}, stemming from the covering of \mathbb{R}^d by balls of radius R centered at \Lambda; for instance, a coarser estimate is \underline{\delta}(\Lambda) \geq \left( \frac{r}{2R} \right)^d.^[13] In geometric terms, Delone sets induce sphere packings in \mathbb{R}^d with minimum inter-center distance $2r, ensuring non-overlapping open balls of radius r, and efficient coverings of \mathbb{R}^d by closed balls of radius R centered at points of \Lambda.^[14] The uniform discreteness also yields a finite intersection property: for any constant c > 0, the open balls of radius c r centered at points of \Lambda intersect only finitely many others, as any intersecting pair of such balls must have centers at most distance $2 c r apart, and the discreteness bounds the number of \Lambda-points in any ball of radius $2 c r by the maximum packing density therein.^[13]

Examples

Periodic Delone Sets

A periodic Delone set is a Delone set \Lambda \subset \mathbb{R}^d that admits translational symmetry under a full-rank lattice \Gamma, meaning \Lambda + \Gamma = \Lambda, where \Gamma is a discrete subgroup of \mathbb{R}^d of rank d.^[10] This periodicity implies that the set of periods \mathrm{per}(\Lambda) = \{ t \in \mathbb{R}^d \mid \Lambda + t = \Lambda \} spans \mathbb{R}^d over \mathbb{R}, making such sets crystallographic in nature.^[10] The primary examples of periodic Delone sets are lattices themselves, which are generated as integer linear combinations of d linearly independent basis vectors \mathbf{b}_1, \dots, \mathbf{b}_d \in \mathbb{R}^d, so \Lambda = \{ \sum_{i=1}^d k_i \mathbf{b}_i \mid k_i \in \mathbb{Z} \}.^[15] A canonical instance is the integer lattice \mathbb{Z}^d in \mathbb{R}^d, where the basis consists of the standard unit vectors; here, the minimal distance between points is 1 (in the \ell_2-norm), yielding a packing radius of $1/2, while the covering radius is \sqrt{d}/2.^[16] For any full-rank lattice \Lambda, the packing radius is half the length of the shortest nonzero vector, and the covering radius is the maximal distance from any point in \mathbb{R}^d to the nearest lattice point.^[17] More generally, periodic Delone sets include multi-lattices, which are finite unions of cosets of a lattice, expressed as \Lambda + P where P is a finite motif subset of the fundamental domain of \Lambda.^[16] For instance, the face-centered cubic (FCC) lattice in \mathbb{R}^3, used to model close-packed crystal structures, can be viewed as a lattice generated by basis vectors such as (1,1,0)/\sqrt{2}, (1,0,1)/\sqrt{2}, and (0,1,1)/\sqrt{2}, achieving a packing density of \pi / (3\sqrt{2}) for sphere packings.^[18] Specific to the periodic case, these sets exhibit exact uniform density given by \mathrm{den}(\Lambda) = 1 / \det(\Gamma) for a lattice \Gamma, or more generally |P| / \det(\Gamma) for a multi-lattice with motif size |P|, where \det(\Gamma) is the volume of the fundamental parallelepiped.^[16] The Voronoi cells—regions D(p) = \{ x \in \mathbb{R}^d \mid \|x - p\| \leq \|x - q\| \ \forall q \in \Lambda \} centered at each point p \in \Lambda—tile \mathbb{R}^d periodically without gaps or overlaps, with each cell having volume equal to \det(\Gamma).^[15] In crystallography, periodic Delone sets model the atomic positions in perfect crystals, where the lattice \Gamma represents the translational symmetries of the Bravais lattice, and multi-lattices account for basis atoms within the unit cell.^[15] This framework underpins the description of crystal structures across the 14 Bravais lattices in three dimensions, linking geometric periodicity to observable diffraction patterns via the dual (reciprocal) lattice.^[15]

Aperiodic Delone Sets

An aperiodic Delone set is a Delone set in Euclidean space that lacks non-trivial translational symmetries, meaning its hull consists entirely of non-periodic configurations with no lattice period.^[10] Such sets often exhibit a pure point diffraction spectrum, distinguishing them from periodic structures while preserving the uniform discreteness and relative density inherent to Delone sets.^[10] They are typically repetitive, ensuring every finite patch appears arbitrarily far away in the set, and may possess finite local complexity, limiting the variety of local configurations up to congruence.^[10] A prominent example is the vertex set of the Penrose tiling in \mathbb{R}^2, which forms a Delone set (scaled such that r \approx 1 and R \approx 2) that is aperiodic despite admitting inflation symmetry via the golden ratio.^[10] The rhombic Penrose tiling enforces aperiodicity through matching rules, such as arrows on tile edges, preventing periodic arrangements while maintaining D_{10} symmetry in its local isomorphism class.^[10] Similarly, the vertex set of the Ammann-Beenker tiling in \mathbb{R}^2 is an aperiodic Delone set with D_8 symmetry and an inflation factor of $1 + \sqrt{2} (the silver ratio), exhibiting quasiperiodic order without translational periodicity.^[10]^[19] In one dimension, the Fibonacci chain provides another illustration: a quasiperiodic Delone set generated by substituting intervals of lengths L and S according to the Fibonacci word, resulting in finite local complexity with prototiles \{L, S\} and no period.^[20] Spiral constructions also yield aperiodic Delone sets in \mathbb{R}^2, such as points on a Fermat spiral given by \{\sqrt{n} e^{i n \alpha} \mid n \in \mathbb{N}\} where the angle \alpha is badly approximable, ensuring uniform discreteness and relative density without periodicity. These sets satisfy Delone parameters bounded by constants related to \alpha's continued fraction approximants, with r \geq 6\sqrt{B} for relative denseness and s \leq \sqrt{C/2} for discreteness. Aperiodic Delone sets model the atomic positions in quasicrystals, such as those observed in Al-Mn alloys discovered in the 1980s, where icosahedral symmetry produces sharp diffraction peaks indicative of long-range order without periodicity.^[21] These structures, first reported in rapidly solidified Al-14 at.% Mn, exemplify how aperiodic Delone configurations underpin real materials with quasiperiodic arrangements.^[21]

Constructions

ε-Nets

An ε-net in \mathbb{R}^d is a set \Lambda such that the balls of radius \varepsilon around points of \Lambda cover \mathbb{R}^d, while the balls of radius \varepsilon/2 around points of \Lambda have disjoint interiors, equivalent to a Delone set with r = \varepsilon and R = \varepsilon. This construction ensures uniform discreteness through the disjoint interiors condition and relative density through the covering property. ε-Nets are also used in finite settings for efficient sampling in high-dimensional data analysis and approximation algorithms.^[22] A standard construction of an ε-net begins with an empty set and iteratively adds points that are at least \varepsilon apart from all existing points using a greedy approach until the set is maximal; this yields a Delone set satisfying the ε-net properties.^[22] The maximality guarantees that the covering radius is at most \varepsilon, while the separation ensures the packing condition. In practice, for finite domains in Euclidean space, algorithmic constructions employ farthest-point sampling, which iteratively selects the point farthest from the current set to build a subset with controlled covering radius, or random sampling refined by removing points closer than \varepsilon to achieve the packing property. These methods approximate the infinite-space greedy construction efficiently, with farthest-point sampling providing good separation and coverage in high dimensions. ε-Nets possess optimal properties in terms of minimal cardinality for achieving a covering of \mathbb{R}^d using balls with disjoint interiors (of radius \varepsilon/2), as the packing condition maximizes space efficiency. Their uniform density is bounded above by $2^d \left( \vol (B(0,1)) \right)^{-1} \varepsilon^{-d} and below by \left( \vol (B(0,1)) \right)^{-1} \varepsilon^{-d}, reflecting near-ideal packing efficiency scaled by the dimension. Every ε-net is a Delone set, but Delone sets generalize the concept to cases where r < R, allowing more flexibility in separation and covering radii; ε-nets represent the tight case where r = R = \varepsilon. In low dimensions, such as \mathbb{R}^1, ε-nets consist of shifts of \varepsilon \mathbb{Z}, where points are spaced exactly \varepsilon apart, achieving both the packing and covering conditions precisely (with actual covering radius \varepsilon/2 \leq \varepsilon).

Model Sets

A model set, also known as a cut-and-project set, is constructed from a higher-dimensional lattice via a projection scheme. Specifically, consider a lattice L in \mathbb{R}^{d+m}, where d is the dimension of the physical space and m the internal space. Let \pi: \mathbb{R}^{d+m} \to \mathbb{R}^d be the orthogonal projection onto the physical space, and let \pi^\perp: \mathbb{R}^{d+m} \to \mathbb{R}^m be the projection onto the internal space. A model set \Lambda is then defined as \Lambda = \{\pi(\gamma) : \gamma \in L \cap (\mathbb{R}^d \times (W + x))\}, where W \subset \mathbb{R}^m is a compact window set with nonempty interior, and x \in \mathbb{R}^m is a shift vector.^[23] If the projection \pi is injective on L (ensuring uniform discreteness) and \pi^\perp(L) is dense in \mathbb{R}^m (ensuring relative density), and W is compact with nonempty interior, then \Lambda is a Delone set. The uniform discreteness parameter r is determined by the minimal distance in L, ensuring no two points in \Lambda are closer than some positive distance, while the relative density parameter R relates to the diameter of W, guaranteeing that every ball of radius R in \mathbb{R}^d contains at least one point of \Lambda.^[23] The construction of a model set proceeds in steps: first, select a lattice L in the embedding space \mathbb{R}^{d+m}, such as the integer lattice \mathbb{Z}^{d+1} for simple cases; second, define the projections \pi and \pi^\perp, often with \pi being a "star map" ensuring density; third, choose a bounded window W in the internal space, for instance, an interval for one-dimensional examples; finally, project the lattice points lying within the "strip" \mathbb{R}^d \times (W + x) onto the physical space. This method naturally produces aperiodic Delone sets when the projections yield irrational rotations or dense subgroups.^[23] A prominent example is the Fibonacci model set in \mathbb{R}^1, obtained from the lattice \mathbb{Z}^2 in \mathbb{R}^2 with an irrational slope \alpha = \frac{1+\sqrt{5}}{2} (the golden ratio). Here, \pi projects onto the line at angle \arctan(\alpha), and W is the interval [0,1) in the orthogonal direction, resulting in a quasiperiodic Delone set whose points correspond to the Beatty sequence \lfloor n\alpha \rfloor, exhibiting the Fibonacci word structure in its differences.^[23] Model sets are closely related to substitution systems, particularly those that are primitive and possess finite local complexity; such systems can generate model sets through their natural extensions, linking algebraic substitutions to geometric projections.^[23] Regular model sets, where the window W has boundary of Lebesgue measure zero, form a subclass of Meyer sets—Delone sets \Lambda such that \Lambda - \Lambda is also Delone (uniformly discrete). These are Delone subsets of \mathbb{Z}-modules and play a key role in the classification of sets with pure point diffraction spectra.^[23]

Applications

Crystallography

Delone sets play a fundamental role in crystallography by modeling the positions of atoms in solid materials, ensuring both discreteness—preventing atomic overlaps through a minimum separation distance r—and relative density, which guarantees a uniform filling of space without large voids.^[24] This dual property captures the essential geometric constraints of crystal structures, where atomic arrangements must be discrete yet space-filling. The concept originated in Boris N. Delone's 1934 work on geometric crystallography, where he introduced (r, R)-systems—now known as Delone sets—to analyze the geometric foundations of crystal lattices and their structural analysis.^[25] In this framework, Delone sets provide a mathematical abstraction for atomic point configurations that satisfy the packing requirements of real crystals. For periodic crystals, Bravais lattices serve as prototypical Delone sets, with the 14 distinct types in three dimensions classifying all possible translationally invariant crystal structures under crystallographic symmetry groups.^[24] For example, the simple cubic Bravais lattice with lattice constant a has Delone parameters r = a (ensuring no two atoms are closer than a) and R = (a \sqrt{3})/2 (the covering radius, radius of the largest empty sphere at the body center); larger R \geq a also works to satisfy r \leq R.^[26] Aperiodic quasicrystals extend this modeling to non-periodic structures, where Delone sets derived from model sets explain sharp diffraction patterns observed in alloys such as icosahedral Al-Pd-Mn.^[25] These quasicrystals, first discovered by Dan Shechtman in 1982 in a rapidly solidified Al-Mn alloy, exhibit long-range order without translational periodicity, with atomic positions forming Delone sets that produce discrete diffraction spectra akin to periodic crystals. Delone sets with finite local complexity (FLC)—meaning only finitely many local configurations (patches) up to translation within any bounded radius—accurately model realistic crystal and quasicrystal structures, as FLC ensures the diffraction spectrum consists of pure point measures, reflecting the sharp Bragg peaks observed experimentally. Voronoi diagrams and Delaunay triangulations, which are dual to Delone sets, are essential computational tools in crystallography for partitioning space around atomic positions and determining nearest-neighbor connectivity, respectively, aiding in the analysis and simulation of crystal structures.^[27]

Coding Theory

Delone sets play a crucial role in coding theory by providing structured sphere packings that form the basis for constellation designs in digital modulation schemes. The uniform discreteness parameter r, the minimal distance between points, ensures that non-overlapping spheres of radius r/2 can be centered at each point in the set, facilitating reliable error separation in the presence of channel noise during signal transmission. This packing property directly translates to the minimum Euclidean distance in the constellation, which determines the error-correcting capability of the code.^[28] Periodic Delone sets, which are lattices, are particularly valuable for constructing high-performance lattice codes. For instance, the Leech lattice in 24 dimensions exemplifies an optimal periodic Delone set, achieving exceptional coding gain through its dense sphere packing; its packing density is tied to a kissing number of 196,560, enabling efficient use of the signal space while minimizing interference. This lattice, derived from binary Golay codes via Construction A, has been instrumental in developing error-correcting codes with superior performance in high-dimensional settings.^[29]^[30] Advanced lattice constructions, such as those based on Construction A* from Delone sets, involve Voronoi cell reduction to optimize the geometry of the lattice basis. This process refines the covering radius R relative to the minimal distance r, maximizing the (r/2)/R ratio to enhance overall code efficiency and reduce decoding complexity. By minimizing the Voronoi cell's irregularity, these constructions yield lattices with improved signal-to-noise ratio performance in practical systems.^[31] In wireless communications, Delone sets underpin key applications, including uncoded quadrature amplitude modulation (QAM), where the square lattice serves as a simple periodic Delone set for constellation points, balancing ease of implementation with adequate error resilience. Sphere decoding algorithms further exploit the Delone structure by searching within bounded Voronoi regions around the received signal, enabling near-maximum-likelihood detection with reduced computational overhead for lattice-based transmissions over fading channels.^[32] The Delone parameters R/(r/2) provide an upper bound on the normalized second moment of the lattice, a critical metric for power efficiency in coded modulation, as lower values indicate better quantization and shaping gains for a given transmit power constraint. This bound ensures that well-conditioned Delone sets yield constellations with minimal average energy while maintaining separation, optimizing the trade-off between rate and reliability.^[28] A representative example is the construction of binary codes from one-dimensional Delone sets, such as the integer lattice \mathbb{Z}, which forms the basis for pulse-amplitude modulation (PAM) schemes; these can be extended to higher dimensions by tensoring with binary error-correcting codes, producing multidimensional lattices suitable for bandwidth-efficient transmission.^[28]

Approximation Algorithms

Delone sets play a key role in approximation algorithms for covering and clustering problems in metric spaces, where they provide structured approximations to optimal solutions by balancing packing and covering properties. In the k-center problem, which seeks to select k points to minimize the maximum distance from any point in the set to the nearest center, an optimal solution forms a Delone set with minimal distance at least 2r (where r is the optimal radius) and covering radius at most r. The greedy farthest-first traversal algorithm constructs a set with minimal distance at least r and covering radius at most 2r, achieving a 2-approximation to the optimal k-center cost.^[33] Delone sets in doubling metrics facilitate low-distortion embeddings into Hilbert space, preserving distances up to a constant factor dependent on the doubling dimension. For a Delone set in a metric with constant doubling dimension, the well-spaced points allow embedding into l_2 (Hilbert space) with O(1) distortion when the dimension is fixed, enabling efficient algorithmic applications like dimension reduction while maintaining geometric properties. A prominent algorithm leveraging Delone sets in doubling metric spaces is the doubling algorithm, which builds a hierarchy of ε-nets—each level forming a Delone set at successively finer scales (e.g., ε, 2ε, 4ε)—to construct spanner graphs with low stretch or support fast nearest neighbor search. This hierarchical structure ensures that queries can be resolved by navigating O(log n) levels, with each Delone set providing a sparse approximation at its scale. In Euclidean space ℝ^d, constructing ε-nets (as (ε, ε)-Delone sets) can be done in O(n log n) time using farthest-point Voronoi diagrams, which efficiently identify the next farthest point in the greedy traversal process. Additionally, random sampling offers a probabilistic approach: selecting O((1/ε)^d log(1/ε)) points uniformly from a bounded region yields a (1+ε)-Delone set with high probability, providing an efficient approximation for covering and packing tasks.

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