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Designer quasicrystals: Cut-and-project sets with pre-assigned properties. (English) Zbl 0982.52018
Baake, Michael (ed.) et al., Directions in mathematical quasicrystals. Providence, RI: AMS, American Mathematical Society. CRM Monogr. Ser. 13, 95-141 (2000).
A cut-and-project set (also called a model set) is a discrete set of the form $ℳ=\left\{{\pi }_{V}\left(l\right)\mid l\in ℒ$, ${\pi }_{W}\left(l\right)\in {\Omega }\right\}$, defined by starting from a Euclidean space ${E}^{N}$ decomposed into a pair of complementary spaces ${E}^{N}=V\oplus W$ with ${\pi }_{V}$ and ${\pi }_{W}$ the associated projections of ${E}^{N}$ onto $V$ and $W$, a lattice $ℒ\subset {E}^{N}$, and a bounded Riemann measurable set ${\Omega }\subset W$, called a window or acceptance domain. The 17 properties concerning a model set (uniformity, diffraction, Ammann bars, symmetry, inflation, local rules, etc.) defined in the first part of the article are useful in quasicrystal modelling, and interesting from a mathematical point of view. The author investigates in detail what conditions on $ℒ$, ${\pi }_{V}$, ${\pi }_{W}$, and ${\Omega }$ in the cut-and-project construction are required for various properties of the resulting model set. Based on a deep understanding of the relation between quasiperiodicity and number theory, the author describes a natural way of obtaining a lattice and projections from a module over an algebraic number field and shows that these necessarily satisfy the conditions for all quasicrystal properties. Some interesting examples and comments concerning the model set symmetries are also included.
##### MSC:
 52C23 Quasicrystals, aperiodic tilings 11P21 Lattice points in specified regions 11R06 Special algebraic numbers 52C07 Lattices and convex bodies in $n$ dimensions 52C22 Tilings in $n$ dimensions (discrete geometry) 82D25 Crystals (statistical mechanics) 11R04 Algebraic numbers; rings of algebraic integers