An extension of algebraic independence of special values for non-lacunary power series

Source abstract

We study the algebraic independence of special values of power series $f(β^{-1})$ , where $β$ is a fixed Pisot or Salem number. In particular, we consider the case where $f(X)=\sum_{n\geq 0} t(n) X^{w(n)}$ is not a lacunary series and is not assumed to satisfy any special functional equation, such as a Mahler-type functional equation. In our main results, we give a new criterion of the algebraic independence of three values. Applying our main results, we prove that the following three values are algebraically independent: $% \begin{gathered} \sum_{n=3}^{\infty}\lfloor n^{y}\rfloorβ^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=3}^{\infty}β^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=1}^{\infty}β^{-\lfloor n^{\log n}\rfloor}, % \end{gathered}$ where $y$ is an arbitrary positive real number. Since our criterion is flexible, we have considerable freedom in choosing the coefficients $(t(n))_{n\geq 0}$ and the exponents $(w(n))_{n\geq 0}$ .