The order type of the set of Pisot numbers
type: Item
instance of: scholarly article
title: The order type of the set of Pisot numbers
zbMATH Open document ID: 0857.11050
DOI: 10.1016/0166-8641(95)00029-1
published in: Topology and its Applications
publication date: 1996-09-22T00:00:00Z
review text: An algebraic integer \(\theta\) is a Pisot (or Pisot-Vijayaraghavan or PV-) number if \(\theta\) is real, \(\theta>1\) and its remaining conjugates lie inside the unit circle. The main result of this paper (answering the question raised by the second author) is the following theorem: The order type of the set \(S\) of all PV-numbers is the ordered sum, \(\sum_{n = 1}^\infty a_n\), where \(a_1 = \omega+1 + \omega^*\) and \(a_{n+1} = a_n \omega+1 + (a_n \omega)^*\) for \(n\geq 1\). The authors are also occupied with some properties of the \(n\)th derived sets \(S^{(n)}\); for example, it is proved that each \(\theta \in S^{(n)}\) \((n \geq 1)\) is a two-sided limit of elements of \(S^{(n-1)}\).
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 54C50
zbMATH DE Number: 896586
zbMATH Keywords: Pisot numbers
zbMATH Keywords: derived sets
zbMATH Keywords: algebraic integers
zbMATH Keywords: order type
zbMATH Keywords: PV-numbers
Wikidata QID: Q56335160
author: David W. Boyd
author: R. Daniel Mauldin
reviewed by: Ladislav Skula
MaRDI profile type: Publication
full work available at URL: 0166-8641(95)00029-1
OpenAlex ID: W2030793812
cites work: Q3907653
cites work: Q4038523
cites work: Q3682570
cites work: Q3480162
cites work: Q5848266
cites work: Ensembles fermés d'entiers algébriques
cites work: Open problems in topology
description: scientific article; zbMATH DE number 896586