On the spectra of Pisot-cyclotomic numbers

We investigate the complex spectra X A ( β ) = ∑ j = 0 n a j β j : n ∈ N , a j ∈ A where β is a quadratic or cubic Pisot-cyclotomic number and the alphabet A is given by 0 along with a finite collection of roots of unity. Such spectra are discrete aperiodic structures with crystallographically forbi...

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Bibliographic Details
Published in:Letters in mathematical physics 2018-07, Vol.108 (7), p.1729-1756
Main Authors: Hare, Kevin G., Masáková, Zuzana, Vávra, Tomáš
Format: Article
Language:English
Subjects:
Complex Systems
Crystallography
Geometry
Group Theory and Generalizations
Mathematical and Computational Physics
Physics
Physics and Astronomy
Quasicrystals
Spectra
Theoretical
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Summary:We investigate the complex spectra X A ( β ) = ∑ j = 0 n a j β j : n ∈ N , a j ∈ A where β is a quadratic or cubic Pisot-cyclotomic number and the alphabet 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.
ISSN:0377-9017
1573-0530
DOI:10.1007/s11005-018-1053-4