Geometric study of the beta-integers for a Perron number and mathematical quasicrystals
type: Item
instance of: scholarly article
title: Geometric study of the beta-integers for a Perron number and mathematical quasicrystals
zbMATH Open document ID: 1075.11007
author: Jean Pierre Gazeau
author: Jean-Louis Verger-Gaugry
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2005-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/51780
review text: 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)].
Mathematics Subject Classification ID: 11A67
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 52C23
zbMATH DE Number: 2184635
zbMATH Keywords: beta-integers
zbMATH Keywords: Perron numbers
zbMATH Keywords: Pisot numbers
zbMATH Keywords: quasi-crystals
zbMATH Keywords: Meyer sets
zbMATH Keywords: Delaunay sets
zbMATH Keywords: \(\beta\)-expansions
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
OpenAlex ID: W1981211389
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DOI: 10.5802/JTNB.437
description: scientific article; zbMATH DE number 2184635