Algebraic independence of the values of power series with unbounded coefficients

Abstract

Many mathematicians have studied the algebraic independence over $\mathbb{Q}$ of the values of gap series, and the values of lacunary series satisfying functional equations of Mahler type. In this paper, we give a new criterion for the algebraic independence over $\mathbb{Q}$ of the values $\sum^{\infty}_{n=0} t(n) \beta^{-n}$ for distinct sequences $(t(n))^{\infty}_{n=0}$ of nonnegative integers, where $\beta$ is a fixed Pisot or Salem number. Our criterion is applicable to certain power series which are not lacunary. Moreover, our criterion does not use functional equations. Consequently, we deduce the algebraic independence of certain values $\sum^{\infty}_{n=0} t_1 (n) \beta^{-n} , \dotsc , \sum^{\infty}_{n=0} t_r( n) \beta^{-n}$ satisfying $$\lim_{n \to \infty , t{i-1} (n) \neq 0} \; \dfrac{t_i(n)}{t_{i-1}(n)^M} = \infty \; (i=2, \dotsc, r)$$ for any positive real number $M$.