21811 articles – 15606 references  [version française]

hal-00014575, version 1

On the complexity of algebraic numbers II. Continued fractions.

Boris Adamczewski () 1, Yann Bugeaud () 2

Abstract: The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. Because of some numerical evidence and a belief that these numbers behave like most numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. More modestly, we may expect that if the sequence of partial quotients of an irrational number $\alpha$ is, in some sense, "simple", then $\alpha$ is either quadratic or transcendental. The term "simple" can of course lead to many interpretations. It may denote real numbers whose continued fraction expansion has some regularity, or can be produced by a simple algorithm (by a simple Turing machine, for example), or arises from a simple dynamical system... The aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity/complexity for the continued fraction expansion of algebraic real numbers of degree at least three.

• 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 – continued fractions

• hal-00014575, version 1
• oai:hal.archives-ouvertes.fr:hal-00014575
• From:
• Submitted on: Monday, 28 November 2005 14:00:13
• Updated on: Wednesday, 1 October 2008 15:04:43