On the complexity of algebraic numbers, II. Continued fractions

By Boris Adamczewski, Yann Bugeaud. 2005

107 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e456a5e5e3792a61a32f876769b1d95ad03aa121

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In Computational Theory and Mathematics

1. Boris Adamczewski
2. Julien Cassaigne
3. Yann Bugeaud
4. Nimish A. Shah

In Mathematical Physics

1. Boris Adamczewski
2. Julien Cassaigne
3. Yann Bugeaud
4. Michel Waldschmidt
5. Samir Siksek

Citations in the corpus

• Diophantine properties of real numbers generated by finite automata (2006, background): “wo distinct positive integers. Let us assume that the sequence a = (an)n≥0 is a non-eventually periodic fixed point for σ. Then, the real numberξ := [a0,a1,a2,...] is transcendental (this is proved in [2]) and is not a U2-number as a consequence of Theorem 7.3. DIOPHANTINE PROPERTIES OF AUTOMATIC REA”
• Dynamics for β-shifts and Diophantine approximation (2007, background): “In particular, we mention the classical ‘Borel conjecture’ on the normality of algebraic irrational numbers (see for instance [2, 7, 13, 24]), and the speculations of Lang [25] on the continued fraction expansion of algebraic numbers of degree at least three (see also [1]).”