Beta-expansions for infinite families of Pisot and Salem numbers
type: Item
instance of: scholarly article
title: Beta-expansions for infinite families of Pisot and Salem numbers
zbMATH Open document ID: 1221.11162
author: Kevin G. Hare
author: David Tweedle
published in: Journal of Number Theory
publication date: 2008-11-18T00:00:00Z
review text: In \(\beta\)-expansions of real numbers the bases \(\beta \in (1,2)\) that are Pisot numbers or Salem numbers play an important role. It is known that if the expansion of \(x \in {\mathbb Q} \cap [0,1)\) is periodic or finite then \(\beta\) must be a Pisot number or a Salem number. The reverse implication for Salem numbers is an open problem. In this paper the authors study some \(\beta\)-expansions and greedy expansions for some special infinite families of Pisot and Salem numbers \(\beta\). For example, \(\beta\)-expansion of the Pisot number defined by \(\Phi_r(x)=x^{r+1}-2x^r+x-1\), \(r \in {\mathbb N}\), is given by \(1^r0^{r-1}1\). The polynomial \(x^k \Phi_r(x)-\Phi_r^{*}(x)\), \(k \in {\mathbb N}\), defines a Salem number. Their greedy expansions are established for \(r \geq k\) and \(r-k\) not divisible by \(2k\). The authors remark that the case \(k=3\), \(r=9\), where \(2k|(r-k)\), leads to an open problem, probably a non-periodic \(\beta\)-expansion.
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 11A63
zbMATH DE Number: 5368241
zbMATH Keywords: beta-expansion
zbMATH Keywords: Salem number
zbMATH Keywords: Pisot number
reviewed by: A. Dubickas
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.02.010
OpenAlex ID: W2002514595
cites work: On univoque Pisot numbers
cites work: Ensembles fermés de nombres algébriques
cites work: Q4135662
cites work: Computational excursions in analysis and number theory
cites work: Pisot and Salem Numbers in Intervals of the Real Line
cites work: Reciprocal Polynomials Having Small Measure
cites work: Pisot Numbers in the Neighborhood of a Limit Point. II
cites work: Pisot numbers in the neighbourhood of a limit point. I
cites work: Reciprocal polynomials having small measure. II
cites work: Q4735248
cites work: On the beta expansion for Salem numbers of degree 6
cites work: Q4377086
cites work: From greedy to lazy expansions and their driving dynamics
cites work: Subexpansions, superexpansions and uniqueness properties in non-integer bases
cites work: Factorization of certain cyclotomic functions
cites work: Q4529547
cites work: Polynomials with small Mahler measure
cites work: On theβ-expansions of real numbers
cites work: Representations for real numbers and their ergodic properties
cites work: Q5332701
cites work: Q3310704
cites work: Algebraic integers whose conjugates lie in the unit circle
DOI: 10.1016/J.JNT.2008.02.010
description: scientific article; zbMATH DE number 5368241