## 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

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

Zbl 1391.11087
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