Algebraic Independence and Mahler's method

TL;DR

This paper advances the understanding of algebraic independence in Mahler's method, providing new results on the independence of values at transcendental points and measures for infinite series, with implications for normality in number theory.

Contribution

It introduces novel results on algebraic independence within Mahler's method, including new measures and examples of normal sets in real numbers.

Findings

Proved algebraic independence of Mahler function values at transcendental points.

Developed new measures of algebraic independence for infinite series.

Provided examples of normal sets in the sense of Chudnovsky for arbitrarily large n.

Abstract

We give some new results on algebraic independence within Mahler's method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence for infinite series of numbers. In particular, our results furnishes, for $n\geq 1$ arbitrarily large, new examples of sets $(\theta_1,...,\theta_n)\in\mrr^n$ normal in the sense of definition formulated by Grigory Chudnovsky (1980).