A certain finiteness property of Pisot number systems
type: Item
instance of: scholarly article
title: A certain finiteness property of Pisot number systems
zbMATH Open document ID: 1052.11055
author: Shigeki Akiyama
author: Hui Rao
author: Wolfgang Steiner
published in: Journal of Number Theory
publication date: 2004-08-06T00:00:00Z
review text: Let \(\beta\) be a real number \(>1\), and let \(\text{Fin}(\beta)\) be the set of nonnegative real numbers having a finite \(\beta\)-expansion. The number \(\beta\) is said to have property (F) if \(\text{Fin}(\beta)=\mathbb Z[1/\beta]\cap \{x\geq 0\}\). This property was introduced by \textit{C. Frougny} and \textit{B. Solomyak} [Ergodic Theory Dyn. Syst. 12, No. 4, 713--723 (1992; Zbl 0814.68065)] who proved in particular that real numbers \(>1\) with property (F) must be Pisot numbers, (the reciprocal is not true). This property is also related to tilings of \(\mathbb R^k\). A weaker property called (W) was introduced by Hollander in his PhD thesis (1996). It reads: (W) \(\forall x\in\mathbb Z[1/\beta]\cap\{x\geq 0\}\) \(\forall\varepsilon > 0\), \(\exists y,z\in\text{Fin}(\beta)\) such that \(x=y-z\) and \(z<\varepsilon\). This property was studied in relation with tilings (instead of property (F)) by the first-named author of this paper [J. Math. Soc. Japan 54, No.~2, 283--308 (2002; Zbl 1032.11033)] and in relation with ergodic results by \textit{N. A. Sidorov} [J. Dyn. Control Syst. 7, No.~4, 447--472 (2002; Zbl 1134.37313), and also Acta Math. Hung. 101, 345--355 (2003; Zbl 1059.28014)]. Sidorov conjectured in particular that only Pisot numbers can have property (W). In the paper under review the authors study property (W) in more detail. We mention in particular their result that numbers with property (W) must be Salem or Pisot, and their algorithmic construction to show (W) or (F) for a given \(\beta\).
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 2083533
zbMATH Keywords: Pisot number
zbMATH Keywords: beta expansion
zbMATH Keywords: weak finiteness
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.02.001
OpenAlex ID: W2018793453
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DOI: 10.1016/J.JNT.2004.02.001
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description: scientific article; zbMATH DE number 2083533