Meyer sets

Meyer sets
Name	Meyer sets
Field	Mathematics; Crystallography
Introduced by	Yves Meyer
Year	1970s

Meyer sets are uniformly discrete point sets in Euclidean space exhibiting both long-range order and aperiodicity; they generalize lattices while appearing in the study of quasicrystals, harmonic analysis, number theory, and mathematical diffraction theory. Originating in work of Yves Meyer and developed alongside research by scholars at institutions such as the Institut des Hautes Études Scientifiques, Massachusetts Institute of Technology, and University of Cambridge, these sets connect to model sets, Pisot–Vijayaraghavan phenomena, and cut-and-project schemes. Their defining mix of algebraic structure and geometric regularity makes them central in investigations tied to Roger Penrose, Alan Mackay, Shechtman, and the broader field of aperiodic order.

Definition and basic properties

A Meyer set is a relatively dense, uniformly discrete subset X of Euclidean space R^n such that X − X is uniformly discrete up to a finite set; equivalently, there exists a finite set F with X − X ⊂ X + F. This definition invokes constructions related to Euclidean space and relies on notions studied by Yves Meyer and practitioners in harmonic analysis and Diophantine approximation. Basic properties include relatively dense packing akin to lattices while permitting nonperiodicity like examples linked to Penrose tiling and Ammann–Beenker tiling. Meyer sets are Meyer’s abstraction of sets arising from algebraic number theory, especially from expansions used by A. Ya. Khinchin, H. Davenport, and Kurt Mahler.

Examples and constructions

Classical examples come from cut-and-project schemes: choose a lattice in R^n × R^m and project a strip (window) from the internal space; this yields model sets that are Meyer sets, as used in constructions by Yves Meyer, J. M. Moody, and R. V. Moody. Quasicrystal examples include the vertex sets of the Penrose tiling, the Ammann–Beenker tiling, and sets obtained from Pisot substitutions related to Salem numbers and Pisot–Vijayaraghavan numbers studied by A. O. Gelfond and Raphaël Salem. Algebraic constructions employ algebraic integers and units from rings of integers in number fields investigated by Emil Artin, Heinz Bauer, and Kurt Hensel. Random perturbations and Meyer sets linked to model sets appear in work at Technion – Israel Institute of Technology and Oak Ridge National Laboratory contexts in mathematical physics.

Algebraic and harmonic analysis

Algebraic aspects involve relations to algebraic number theory, where Meyer sets arise from projection of lattices associated to rings of integers in algebraic number fields like cyclotomic fields studied by Carl Friedrich Gauss and Leopold Kronecker. Harmonic analysis on Meyer sets studies Fourier transform properties, Fourier–Bohr coefficients, and almost periodic measures linked to research by Harald Bohr, Norbert Wiener, and Jean Bourgain. The autocorrelation and diffraction measures for Meyer sets often exhibit pure point spectra tied to dual modules and dual groups appearing in the work of Hillel Furstenberg, John von Neumann, and Israel Gelfand.

Dynamical and diffraction properties

The dynamical systems formed by translation actions on the hull of a Meyer set connect to topological dynamics and ergodic theory studied by Furstenberg, Anatole Katok, and Michael Herman. Minimality, unique ergodicity, and eigenvalue structure for the translation action reflect the pure point diffraction seen in ideal model sets, as analyzed by D. Lenz, Robinson and J. Kellendonk. Diffraction patterns of Meyer sets produce Bragg peaks and singular continuous components in diffraction spectra analogous to observations by Daniel Shechtman and experimental teams in material science labs studying alloy quasicrystals. The link between dynamical eigenfunctions and diffraction intensities uses methods from Alexandre Beurling and Harald Bohr.

Connections to quasicrystals and aperiodic order

Meyer sets provide a mathematical framework for aperiodic order and quasicrystalline structures discovered experimentally by Daniel Shechtman and theoretically modeled by Roger Penrose and Alan Mackay. Model sets arising from cut-and-project methods give idealized atomic positions for quasicrystals such as icosahedral phases related to studies at National Institute of Standards and Technology and Max Planck Institute for Solid State Research. Meyer sets formalize long-range aperiodic order without translational symmetry and connect to tiling spaces like those in the Penrose tiling and substitution systems analyzed by John Conway and N. G. de Bruijn.

Classification and characterization theorems

Characterization theorems identify Meyer sets as model sets under conditions such as regularity of windows, pure point diffraction, and almost periodicity; these results were advanced by Yves Meyer, R.V. Moody, J.-B. Gouéré, and Daniel Lenz. Equivalence criteria relate the finite local complexity, repetitivity, and Meyer property to algebraic conditions on difference sets and embedding into lattices of higher-rank groups studied by Hermann Weyl and E. Artin. Classification efforts also involve cohomological invariants and K-theory techniques drawn from work by Alain Connes and Nigel Higson.

Applications and open problems

Applications include modeling atomic structure in materials science, designing photonic quasicrystals explored at Harvard University and MIT, and coding theory links to cryptography and signal processing in laboratories at Bell Labs. Open problems center on complete classification of Meyer sets in higher dimensions, criteria for pure point versus mixed spectral types as pursued by J.-B. Gouéré and Lorenzo Sadun, and connections between Meyer sets and deeper algebraic structures in arithmetic geometry and ergodic theory. Current research programs at institutions like CNRS, ETH Zurich, and Australian National University continue to investigate stability under perturbations, random tilings, and computational recognition of Meyer properties.

Category:Mathematical structures