  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.

Number Theory