Open Access
June 2019 Arithmetical properties of real numbers related to beta-expansions
Hajime Kaneko
Funct. Approx. Comment. Math. 60(2): 195-226 (June 2019). DOI: 10.7169/facm/1714


The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} \beta^{-w(m)}\), where \(\beta\) is a fixed Pisot or Salem number and \(w(m)\) (\(m=0,1,\ldots\)) are distinct sequences of nonnegative integers with \(w(m+1)>w(m)\) for any sufficiently large \(m\). We first introduce the algebraic independence results of such values. Our results are applicable to certain sequences \(w(m)\) (\(m=0,1,\ldots\)) with \(\lim_{m\to\infty}w(m+1)/w(m)=1.\) For example, we prove that two numbers \[\sum_{m=1}^{\infty}\beta^{-\lfloor \varphi(m)\rfloor}, \quad \sum_{m=3}^{\infty}\beta^{-\lfloor a(m)\rfloor}\] are algebraically independent, where \(\varphi(m)=m^{\log m}\) and \(a(m)=m^{\log\log m}\). Moreover, we also give the linear independence results of real numbers. Our results are applicable to the values \(\sum_{m=0}^{\infty}\beta^{-\lfloor m^\rho\rfloor}\), where \(\beta\) is a Pisot or Salem number and \(\rho\) is a real number greater than 1.


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Hajime Kaneko. "Arithmetical properties of real numbers related to beta-expansions." Funct. Approx. Comment. Math. 60 (2) 195 - 226, June 2019.


Published: June 2019
First available in Project Euclid: 28 March 2018

zbMATH: 07068531
MathSciNet: MR3964260
Digital Object Identifier: 10.7169/facm/1714

Primary: 11J91
Secondary: 11J72 , 11K16

Keywords: algebraic independence , beta expansion , Pisot Numbers , Power series , Salem numbers

Rights: Copyright © 2019 Adam Mickiewicz University

Vol.60 • No. 2 • June 2019
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