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Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. (English) Zbl 1075.11007

In this dense paper the authors study genetic properties of the set \(\mathbb Z_\beta\) of \(\beta\)-integers for \(\beta\) a Perron number (\(\beta\)-integers are real numbers that are equal to the integer part of their \(\beta\)-expansion). They prove in particular that these sets can be obtained by two cut-and-project schemes. When \(\beta\) is a Pisot number, they obtain a new proof that \(\mathbb Z_\beta\) is a Meyer set. Among other results they also give a link with Lagarias’ classification of Delaunay sets.
Note that the reference [MVG] appeared with a slightly different title in [G. Muraz and J.-L. Verger-Gaugry, Exp. Math. 14, No. 1, 47–57 (2005; Zbl 1108.52021)].

MSC:

11A67 Other number representations
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
52C23 Quasicrystals and aperiodic tilings in discrete geometry

Citations:

Zbl 1108.52021
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References:

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