Comments on the spectra of Pisot numbers
type: Item
instance of: scholarly article
title: Comments on the spectra of Pisot numbers
zbMATH Open document ID: 1132.11053
author: David Garth
author: Kevin G. Hare
published in: Journal of Number Theory
publication date: 2007-01-26T00:00:00Z
review text: For \(q\in (1,2)\) and an integer \(m\geq 1,\) write \(Y^m(q)=\{y_0=0< y_1<y_2<y_3<\dots\},\) where each \(y_i\) is expressible by a (finite) sum \(\sum_{k=0}^{\infty} u_k q^k\) with \(u_0,u_1, u_2,\dots\in \{0,\dots,m\}.\) The set \(Y^m(q)\) is called the spectra of \(q.\) The authors continue the study of the set \(Y^m(q)\) and the gap sequence \(y_1-y_0, y_2-y_1, y_3-y_2, \dots\) for Pisot numbers \(q.\) (It is known that \(q\) is a Pisot number, if and only if, \(\liminf_{k\to \infty} (y_{k+1}-y_k) >0\) for all \(m \geq 1\).) In particular, they show that if the gap \(\beta \in \{y_n-y_{n-1} : y_n \in Y^m(q)\}\}\) occurs with frequency \(\alpha\) in \(Y^m(q)\) then \(\alpha\) must be a polynomial in \(q\) with rational coefficients. Suppose \(q\) is a quadratic Pisot number, which the root of \(x^2-ax+b=0\) and \(m=[q].\) They compute the frequencies of \(1, q-a+1\) and \(a-q\) in \(Y^m(q).\)
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 5118758
zbMATH Keywords: Pisot numbers
zbMATH Keywords: spectra
zbMATH Keywords: substitution algebra
zbMATH Keywords: frequency
zbMATH Keywords: gaps
reviewed by: A. Dubickas
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.02.003
OpenAlex ID: W2039338734
cites work: Automatic Sequences
cites work: Q4101884
cites work: On algebraic equations with all but one root in the interior of the unit circle. To my teacher and former colleague Erhard Schmidt on his 75th birthday
cites work: On a property of Pisot numbers and related questions
cites work: On the sequence of numbers of the form \(q^{n_1} + \cdots + q^{n_k}\)
cites work: Characterization of the unique expansions $1=\sum^{\infty}_{i=1}q^{-n_ i}$ and related problems
cites work: On the sequence of numbers of the form $ε₀ + ε₁q + ... + ε_nq^n$, $ε_i ∈ {0,1}$
cites work: Q4370884
cites work: On a problem of Tamas Varga
cites work: Developments in non-integer bases
cites work: A property of Pisot numbers
cites work: Finite beta-expansions
cites work: The structure of the spectra of Pisot numbers.
cites work: Matrix Analysis
cites work: On digit expansions with respect to linear recurrences
cites work: Substitution dynamical systems - spectral analysis
cites work: Q5526189
DOI: 10.1016/J.JNT.2006.02.003
description: scientific article; zbMATH DE number 5118758