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On the spectra of Pisot-cyclotomic numbers. (English) Zbl 1395.52026

Summary: We investigate the complex spectra \[ X^{\mathcal A} (\beta)=\left\{ \sum_{j=0}^n a_j\beta^j : n\in \mathbb N,\, a_j\in \mathcal A\right\} \] 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.

MSC:

52C23 Quasicrystals and aperiodic tilings in discrete geometry
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11A63 Radix representation; digital problems
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