Meyer's concept of quasicrystal and quasiregular sets

AbstractThis paper relates two mathematical concepts of long-range order of a set of atoms Λ, each of which is based on restrictions on the set of interatomic distances Λ−Λ. A set Λ in ℝn is aMeyer set if Λ is a Delone set and there is a finite setF such that $$\Lambda - \Lambda \subseteq \Lambda + F.{\text{ Y}}$$ . Meyer proposed that such sets include the possible structures of quasicrystals. He obtained a structure theory for such sets, which reformulates results that he obtained in harmonic analysis around 1970, and which relates these sets to cut-and-project sets. In geometric crystallography V.I. Galiulin introduced the concept ofquasiregular set, which is a set Λ such that both Λ and Λ−Λ are Delone sets. This paper shows that these two concepts are identical.

[1]  Charles Radin,et al.  Space tilings and substitutions , 1995 .

[2]  Peter Kramer,et al.  Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells , 1982 .

[3]  Le Tu Quoc Thang,et al.  The geometry of quasicrystals , 1993 .

[4]  P. Steinhardt,et al.  Quasicrystals: a new class of ordered structures , 1984 .

[5]  A. Hof.  Quasicrystals, aperiodicity and lattice systems , 1992 .

[6]  E. Arthur Robinson,et al.  Ergodic Theory of ℤ d Actions: The dynamical theory of tilings and Quasicrystallography , 1996 .

[7]  J. Cahn,et al.  Metallic Phase with Long-Range Orientational Order and No Translational Symmetry , 1984 .

[8]  Yves Meyer,et al.  Quasicrystals, Diophantine approximation and algebraic numbers , 1995 .

[9]  C. Radin.  Global order from local sources , 1991 .

[10]  B. Delaunay.  Neue Darstellung der geometrischen Kristallographie , 1933 .

[11]  P. Kramer,et al.  On Periodic and Non-periodic Space Fillings of E , 1984 .

[12]  R. Moody.  Meyer Sets and the Finite Generation of Quasicrystals , 1995 .

[13]  A. Katz,et al.  Theory of matching rules for the 3-dimensional Penrose tilings , 1988 .

[14]  Yves Meyer,et al.  Algebraic numbers and harmonic analysis , 1972 .

[15]  Charles Radin,et al.  The pinwheel tilings of the plane , 1994 .

[16]  Enrico Bombieri,et al.  Which distributions of matter diffract? An initial investigation , 1986 .

[17]  J. Socolar.  Weak matching rules for quasicrystals , 1990 .

[18]  At Hof,et al.  On diffraction by aperiodic structures , 1995 .

[19]  L. Levitov.  Local rules for quasicrystals , 1988 .

[20]  J. Rhyner,et al.  Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings , 1986 .

[21]  Peter Engel,et al.  Geometric Crystallography: An Axiomatic Introduction to Crystallography , 1986 .

[22]  V. Elser.  The diffraction pattern of projected structures , 1986 .

[23]  M. Senechal.  Quasicrystals and geometry , 1995 .

[24]  Jiri Patera.  Noncrystallographic Root Systems and Quasicrystals , 1997 .

[25]  de Ng Dick Bruijn,et al.  Quasicrystals and their Fourier transform , 1986 .

[26]  Le Tu Quoc Thang.  Local rules for pentagonal quasi-crystals , 1995, Discret. Comput. Geom..

[27]  Y. Meyer,et al.  Nombres de Pisot, Nombres de Salem et Analyse Harmonique , 1970 .

[28]  Barry Simon,et al.  Singular continuous spectrum for palindromic Schrödinger operators , 1995 .

[29]  S. Burkov.  Absence of weak local rules for the planar quasicrystalline tiling with the 8-fold rotational symmetry , 1988 .

[30]  A. Hof,et al.  Diffraction by aperiodic structures at high temperatures , 1995 .

[31]  N. D. Bruijn.  Algebraic theory of Penrose''s non-periodic tilings , 1981 .

[32]  C. Janot,et al.  Quasicrystals: A Primer , 1992 .