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

## Abstract

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.

## Citation

Hajime Kaneko. "Arithmetical properties of real numbers related to beta-expansions." Funct. Approx. Comment. Math. 60 (2) 195 - 226, June 2019. https://doi.org/10.7169/facm/1714

## Information

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

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

Subjects:
Primary: 11J91
Secondary: 11J72 , 11K16

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