In mathematics, a Meyer set or almost lattice is a set relatively dense ''X'' of points in the Euclidean plane or a higher-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

such that its Minkowski difference with itself is uniformly discrete. Meyer sets have several equivalent characterizations; they are named after

Yves Meyer Yves F. Meyer (; born 19 July 1939) is a French mathematician. He is among the progenitors of wavelet theory, having proposed the Meyer wavelet. Meyer was awarded the Abel Prize in 2017. Biography Born in Paris to a Jewish family, Yves Meyer ...

, who introduced and studied them in the context of diophantine approximation. Nowadays Meyer sets are best known as mathematical model for

quasicrystal A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical ...

s. However, Meyer's work precedes the discovery of quasicrystals by more than a decade and was entirely motivated by number theoretic questions..

Definition and characterizations

A subset ''X'' of a

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

is relatively dense if there exists a number ''r'' such that all points of ''X'' are within distance ''r'' of ''X'', and it is uniformly discrete if there exists a number ''ε'' such that no two points of ''X'' are within distance ''ε'' of each other. A set that is both relatively dense and uniformly discrete is called a Delone set. When ''X'' is a subset of a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

, its Minkowski difference ''X'' − ''X'' is the set of differences of pairs of elements of ''X''. With these definitions, a Meyer set may be defined as a relatively dense set ''X'' for which ''X'' − ''X'' is uniformly discrete. Equivalently, it is a Delone set for which ''X'' − ''X'' is Delone, or a Delone set ''X'' for which there exists a finite set ''F'' with ''X'' − ''X'' ⊂ ''X'' + ''F'', Section 7. Some additional equivalent characterizations involve the set :

X^\epsilon = \

defined for a given ''X'' and ''ε'', and approximating (as ''ε'' approaches zero) the definition of the

reciprocal lattice In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...

of a

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

. A relatively dense set ''X'' is a Meyer set if and only if * For all ''ε'' > 0, ''X''^''ε'' is relatively dense, or equivalently * There exists an ''ε'' with 0 < ''ε'' < 1/2 for which ''X''^''ε'' is relatively dense. A

character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...

of an additively closed subset of a vector space is a function that maps the set to the unit circle in the plane of

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s, such that the sum of any two elements is mapped to the product of their images. A set ''X'' is a

harmonious set In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters. Equivalently, a suitably defined dual set is relatively dense in the Pontryagin dual o ...

if, for every character ''χ'' on the additive closure of ''X'' and every ''ε'' > 0, there exists a continuous character on the whole space that ''ε''-approximates ''χ''. Then a relatively dense set ''X'' is a Meyer set if and only if it is harmonious.

Examples

Meyer sets include *The points of any

*The vertices of any rhombic

Penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without ...

*The

Minkowski sum In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...

of another Meyer set with any nonempty

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. T ...

*Any relatively dense subset of another Meyer set, Corollary 6.7.

References

{{reflist Metric geometry Crystallography Lattice points