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


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.


Boris Adamczewski

Département Mathématiques, CNRS, Université Claude Bernard Lyon 1, 69622 Villeurbanne, France

Yann Bugeaud

L'UFR de Mathématique et d'Informatique, Université Louis Pasteur, 67084 Strasbourg, France