On the index of Siegel grids and its application to the tomography of quasicrystals

We give a characterization of when the index of Siegel grids is finite. As a main application, we solve a basic decomposition problem for the discrete tomography of quasicrystals that live on finitely generated Z-modules in some R^s.

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

[3]  Peter Schwander,et al.  An approach to quantitative high-resolution transmission electron microscopy of crystalline materials , 1995 .

[4]  R. Tennant.  Algebra , 1941, Nature.

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

[6]  M. Baake,et al.  Discrete tomography of Penrose model sets , 2006, math-ph/0610056.

[7]  G. Herman,et al.  Discrete tomography : foundations, algorithms, and applications , 1999 .

[8]  Kim,et al.  Mapping projected potential, interfacial roughness, and composition in general crystalline solids by quantitative transmission electron microscopy. , 1993, Physical review letters.

[9]  G. Herman,et al.  Advances in discrete tomography and its applications , 2007 .

[10]  M. Schreiber,et al.  Quasicrystals : an introduction to structure, physical properties, and applications , 2002 .

[11]  R. Moody.  Meyer Sets and Their Duals , 1997 .

[12]  Michael Baake,et al.  A Guide to Mathematical Quasicrystals , 2002 .

[13]  P. Kramer,et al.  Projection of the Danzer tiling , 1994 .

[14]  C. Siegel,et al.  Lectures on the Geometry of Numbers , 1989 .

[15]  Robert V. Moody,et al.  Model Sets: A Survey , 2000 .

[16]  Peter Gritzmann,et al.  On the Reconstruction of Finite Lattice Sets from their X-Rays , 1997, DGCI.

[17]  M. Baake,et al.  Weighted Dirac combs with pure point diffraction , 2002, math/0203030.

[18]  Ludwig Danzer,et al.  Three-dimensional analogs of the planar penrose tilings and quasicrystals , 1989, Discret. Math..

[19]  L. Danzer.  Quasiperiodicity: Local and global aspects , 1991 .

[20]  O. Bolotina,et al.  On stability of the , 2003 .

[21]  Peter Gritzmann,et al.  On Stability, Error Correction, and Noise Compensation in Discrete Tomography , 2006, SIAM J. Discret. Math..

[22]  L. Washington.  Introduction to Cyclotomic Fields , 1982 .

[23]  John W. Cahn,et al.  Quasicrystals , 2001, Journal of research of the National Institute of Standards and Technology.

[24]  Levine,et al.  Quasicrystals with arbitrary orientational symmetry. , 1985, Physical Review B (Condensed Matter).

[25]  Peter Gritzmann,et al.  Discrete tomography of planar model sets. , 2006, Acta crystallographica. Section A, Foundations of crystallography.

[26]  A. Keller.  Mathematical and Physical Sciences , 1933, Nature.

[27]  M. Baake,et al.  Generalized model sets and dynamical systems , 2000 .

[28]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[29]  M. Schlottmann,et al.  Cut-and-project sets in locally compact Abelian groups , 1998 .

[30]  Attila Kuba,et al.  Advances in Discrete Tomography and Its Applications (Applied and Numerical Harmonic Analysis) , 2007 .

[31]  Sven de Vries,et al.  Reconstructing Crystalline Structures from Few Images Under High Resolution Transmission Electron Microscopy , 2003 .

[32]  G. Birkhoff,et al.  A survey of modern algebra , 1942 .

[33]  W. Jäger,et al.  Mathematics – key technology for the future , 2003 .

[34]  Garrett Birkhoff,et al.  A survey of modern algebra , 1942 .

[35]  K.-P Nischke,et al.  A construction of inflation rules based onn-fold symmetry , 1996, Discret. Comput. Geom..