Quasicrystals, tilings, and algebraic number theory: some preliminary connections.

*(English)*Zbl 0617.43002
The legacy of Sonya Kovalevskaya, Proc. Symp., Radcliffe Coll. 1985, Contemp. Math. 64, 241-264 (1987).

[For the entire collection see Zbl 0601.00007.]

The notion of ”quasicrystals” has its origin in physics: the essential property of physical quasicristals is that their diffraction pattern consists of spots, but the pattern is not compatible with a translationally periodic arrangement of atoms. Since the diffraction patterns are given by the Fourier transform of the density of the diffraction material the mathematical characterization of quasicrystals is given by conditions on both the density \(\rho\) and its Fourier transform \({\hat \rho}\). A density \(\rho\) is called a quasicrystal if it has the properties of atomic density, atmonic diffusion and homogeneity. [Compare J. W. Cahn and the second author, An introduction to quasicrystals, ibid. 64, 265-286 (1987) for an introduction.]

In this paper there are given some results on quasicrystals in dimension 1, exploring some of the relationships between tilings defined by substitution rules, quasi-periodicity, and number theory. The reader should take notice of the fact that pages 243 and 244 were interchanged in print.

The notion of ”quasicrystals” has its origin in physics: the essential property of physical quasicristals is that their diffraction pattern consists of spots, but the pattern is not compatible with a translationally periodic arrangement of atoms. Since the diffraction patterns are given by the Fourier transform of the density of the diffraction material the mathematical characterization of quasicrystals is given by conditions on both the density \(\rho\) and its Fourier transform \({\hat \rho}\). A density \(\rho\) is called a quasicrystal if it has the properties of atomic density, atmonic diffusion and homogeneity. [Compare J. W. Cahn and the second author, An introduction to quasicrystals, ibid. 64, 265-286 (1987) for an introduction.]

In this paper there are given some results on quasicrystals in dimension 1, exploring some of the relationships between tilings defined by substitution rules, quasi-periodicity, and number theory. The reader should take notice of the fact that pages 243 and 244 were interchanged in print.

Reviewer: P.Kirschenhofer

##### MSC:

43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |

11L03 | Trigonometric and exponential sums, general |

82D25 | Statistical mechanical studies of crystals |

11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |

11B37 | Recurrences |

11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |

52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |