Arithmetical properties of real numbers related to beta-expansions

Abstract

The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} β^{-w(m)}\), where \(β\) 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 criteria for the algebraic independence of such values. Our criteria 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 \[\sum_{m=1}^{\infty}β^{-\lfloor φ(1,0;m)\rfloor}, \sum_{m=3}^{\infty}β^{-\lfloor φ(0,1;m)\rfloor}\] are algebraically independent, where \(φ(1,0;m)=m^{\log m}\) and \(φ(0,1;m)=m^{\log\log m}\). \par Moreover, we also give criteria for linear independence of real numbers. Our criteria are applicable to the values \(\sum_{m=0}^{\infty}β^{-\lfloor m^ρ\rfloor}\), where \(β\) is a Pisot or Salem number and \(ρ\) is a real number greater than 1.