Arithmetical properties of real numbers related to beta-expansions

Abstract

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