On the complexity of algebraic numbers I. Expansions in integer bases

Abstract

Let $b \ge 2$ be an integer. We prove that the $b$-ary expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.