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hal-00014567, version 1

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

Boris Adamczewski () 1, Yann Bugeaud () 2

Abstract: Let $b \ge 2$ be an integer. We prove that the $b$-adic 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.

  • 1:  Institut Camille Jordan (ICJ)
  • CNRS : UMR5208 – Université Claude Bernard - Lyon I – Ecole Centrale de Lyon – Institut National des Sciences Appliquées (INSA) - Lyon
  • 2:  Institut de Recherche Mathématique Avancée (IRMA)
  • CNRS : UMR7501 – Université Louis Pasteur - Strasbourg I
  • Domain : Mathematics/Number Theory
  • Keywords : Transcendental numbers – $b$-adic expansion – finite automata – Subspace theorem
 
  • hal-00014567, version 1
  • http://hal.archives-ouvertes.fr/hal-00014567
  • oai:hal.archives-ouvertes.fr:hal-00014567
  • From: Boris Adamczewski <>
  • Submitted on: Monday, 28 November 2005 12:17:47
  • Updated on: Wednesday, 1 October 2008 15:05:25