Mahler method: linear relations, transcendence and applications to automatic numbers. (Méthode de Mahler: relations linéaires, transcendance et applications aux nombres automatiques.) (French. English summary) Zbl 1440.11132

Adamczewski, Boris; Faverjon, Colin

Proc. Lond. Math. Soc. (3) 115, No. 1, 55-90 (2017).

Summary: This paper is devoted to the so-called Mahler method. We precisely describe the structure of linear relations between values at algebraic points of Mahler functions. Given a number field \(k\), a Mahler function \(f(z)\in k\{z\}\), and an algebraic number \(\alpha, 0<|\alpha|<1\), that is not a pole of \(f\), we show that one can always determine whether the number \(f(\alpha)\) is transcendental or not. In the latter case, we further obtain that \(f(\alpha)\) belongs to the number field \(k(\alpha)\). We also discuss some consequences of this theory concerning a classical number theoretical problem: the study of the sequence of digits of the expansion of algebraic numbers in integer bases, or, more generally in algebraic bases. Our results are derived from a recent theorem of P. Philippon [J. Lond. Math. Soc., II. Ser. 92, No. 3, 596–614 (2015; Zbl 1391.11087)] that we refine. We also simplify its proof.

MSC:

11J81	Transcendence (general theory)
11J85	Algebraic independence; Gel’fond’s method
11B85	Automata sequences

Keywords:

automatic numbers; Mahler method

Citations:

Zbl 1391.11087