9,854 research outputs found
On the complexity of algebraic numbers II. Continued fractions
The continued fraction expansion of an irrational number is
eventually periodic if and only if 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 is, in some sense, "simple", then
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
On the Maillet--Baker continued fractions
We use the Schmidt Subspace Theorem to establish the transcendence of a class
of quasi-periodic continued fractions. This improves earlier works of Maillet
and of A. Baker. We also improve an old result of Davenport and Roth on the
rate of increase of the denominators of the convergents to any real algebraic
number
Structured matrices, continued fractions, and root localization of polynomials
We give a detailed account of various connections between several classes of
objects: Hankel, Hurwitz, Toeplitz, Vandermonde and other structured matrices,
Stietjes and Jacobi-type continued fractions, Cauchy indices, moment problems,
total positivity, and root localization of univariate polynomials. Along with a
survey of many classical facts, we provide a number of new results.Comment: 79 pages; new material added to the Introductio
Euclidean algorithms are Gaussian
This study provides new results about the probabilistic behaviour of a class
of Euclidean algorithms: the asymptotic distribution of a whole class of
cost-parameters associated to these algorithms is normal. For the cost
corresponding to the number of steps Hensley already has proved a Local Limit
Theorem; we give a new proof, and extend his result to other euclidean
algorithms and to a large class of digit costs, obtaining a faster, optimal,
rate of convergence. The paper is based on the dynamical systems methodology,
and the main tool is the transfer operator. In particular, we use recent
results of Dolgopyat.Comment: fourth revised version - 2 figures - the strict convexity condition
used has been clarifie
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