A new criterion for algebraic independence of special values of power series at Pisot or Salem numbers

Abstract: In this talk, we introduce a new criterion for the algebraic independence of special values of power series at Pisot or Salem numbers. More precisely, we derive the algebraic independence of the following three values: \[ \sum_{n\geq0} t_1(n)\beta^{-w_1(n)},\quad \sum_{n\geq0} t_2(n)\beta^{-w_1(n)},\quad \sum_{n\geq0} t_3(n)\beta^{-w_2(n)}, \] where $(t_1(n))_{n\geq0}, (t_2(n))_{n\geq0}, (t_3(n))_{n\geq0}$ are sequences of positive integers, $(w_1(n))_{n\geq0}, (w_2(n))_{n\geq0}$ are eventually strictly increasing sequences of nonnegative integers, and $\beta$ is a Pisot or Salem number.

For instance, we prove that the following three values are algebraically independent: \[ \sum_{n=3}^{\infty}\lfloor n^{y}\rfloor\beta^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=3}^{\infty}\beta^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=1}^{\infty}\beta^{-\lfloor n^{\log n}\rfloor}, \] where $y$ is an arbitrary positive real number. Our criterion is flexible because it can also be used to prove the algebraic independence without using functional equations. This is joint work with Hajime Kaneko.

dynamical systemsnumber theory

Audience: researchers in the topic

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