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

TL;DR

This paper advances the understanding of Mahler functions by establishing new algebraic independence results and measures, including at transcendental points, with applications to Mahler numbers and normality in number theory.

Contribution

It introduces novel measures of algebraic independence for Mahler functions at transcendental points and applies these to classify Mahler numbers and construct normal sets.

Findings

Proves Mahler numbers are not in class U.

Provides new measures of algebraic independence.

Constructs examples of normal sets in real numbers.

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).