Arithmetical properties of real numbers related to beta-expansions

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.