Geometric study of the beta-integers for a Perron number and mathematical quasicrystals
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 [\textit{G. Muraz} and \textit{J.-L. Verger-Gaugry}, Exp. Math. 14, No. 1, 47--57 (2005; Zbl 1108.52021)].
scientific article; zbMATH DE number 2184635
scientific article