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On the spectra of Pisot-cyclotomic numbers

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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.

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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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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Kevin G. Hare

  2. Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00, Praha 2, Czech Republic

    Zuzana Masáková

  3. Department of Algebra, FMP, Charles University, Sokolovská 83, 18600, Praha 8, Czech Republic

    Tomáš Vávra

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  1. Kevin G. Hare
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Correspondence to Tomáš Vávra.

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Hare, K.G., Masáková, Z. & Vávra, T. On the spectra of Pisot-cyclotomic numbers. Lett Math Phys 108, 1729–1756 (2018). https://doi.org/10.1007/s11005-018-1053-4

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  • DOI: https://doi.org/10.1007/s11005-018-1053-4

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