Mahler's method: linear relations, transcendence and applications to the automatic numbers

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)k{z}, and an algebraic number , 0<||<1, that is not a pole of f, we show that one can always determine whether the number f() is transcendental or not. In the latter case, we further obtain that f() belongs to the number field k(). 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 Philippon [Groupes de Galois et nombres automatiques', J. Lond. Math. Soc. 95 (2015) 596-614] that we refine. We also simplify its proof.