Algebraic independence and normality of the values of Mahler's functions

Abstract: The main purpose of this article is to provide new results on algebraic independence of values of Mahler functions and their generalizations. Simultaneously, we establish new measures of algebraic independence for these values. Among the other things, we provide a measure of algebraic independence for values of Mahler's functions at complex transcendental points, a result of type which has never appeared in the literature in the past. As an example of application of our new measures of algebraic independence, we prove that a Mahler number does not belong to the class $U$ in Mahler's classification. Also, our results imply new examples, for $n\geq 1$ arbitrarily large, of sets $\left(\theta_1,\dots,\theta_n\right)\in\mathbb{R}^n$ normal in the sense of G.~Chudnovsky (1980).