Arithmetical properties of real numbers related to beta-expansions (1708.03093v1)

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.