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Letters in Mathematical Physics

, Volume 108, Issue 7, pp 1729–1756 | Cite as

On the spectra of Pisot-cyclotomic numbers

  • Kevin G. Hare
  • Zuzana Masáková
  • Tomáš Vávra
Article
  • 88 Downloads

Abstract

We investigate the complex spectra
$$\begin{aligned} X^\mathcal A(\beta )=\left\{ \sum _{j=0}^na_j\beta ^j : n\in \mathbb N,\ a_j\in \mathcal A\right\} \end{aligned}$$
where \(\beta \) is a quadratic or cubic Pisot-cyclotomic number and the alphabet \(\mathcal A\) is given by 0 along with a finite collection of roots of unity. Such spectra are discrete aperiodic structures with crystallographically forbidden symmetries. We discuss in general terms under which conditions they possess the Delone property required for point sets modeling quasicrystals. We study the corresponding Voronoi tilings and we relate these structures to quasilattices arising from the cut-and-project method.

Keywords

Pisot-cyclotomic number Voronoi tiling Cut-and-project Quasicrystals 

Mathematics Subject Classification

52C23 11K16 11A63 

Notes

Acknowledgements

This work was supported by the Czech Science Foundation, Grant No. 13-03538S. We also acknowledge financial support of the Grant Agency of the Czech Technical University in Prague, Grant No. SGS17/193/OHK4/3T/14. The Research of K. G. Hare was supported by NSERC Grant RGPIN-2014-03154.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of MathematicsFNSPE, Czech Technical University in PraguePraha 2Czech Republic
  3. 3.Department of Algebra, FMPCharles UniversityPraha 8Czech Republic

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