Adamczewski Boris & Yann Bugeaud
On the complexity of algebraic numbers II. Continued fractions.
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.
