Diophantine conditions in small divisors and transcendental number theory

E. Muñoz Garcia; R. Pérez-Marco

doi:10.3934/dcds.2003.9.1401

Article Contents

2003, Volume 9, Issue 6: 1401-1409. Doi: 10.3934/dcds.2003.9.1401

This issue Previous Article Dispersive estimate for the wave equation with the inverse-square potential Next Article Noncommutative dynamical systems with two generators and their applications in analysis

Diophantine conditions in small divisors and transcendental number theory

E. Muñoz Garcia^1, and
R. Pérez-Marco^2,

1.
Institut des Hautes Études Scientifiques, 35, Route de Chartres, 91440-Bures-Sur-Yvette
2.
University of California Los Angeles, 520 Portola Plaza, Los Angeles, California 90095

Received: September 30, 2002

Revised: April 30, 2003

Published: October 2003

Abstract Related Papers Cited by

Abstract

We present analogies between Diophantine conditions appearing in the theory of Small Divisors and classical Transcendental Number Theory. Let K be a number field. Using Bertrand's postulate, we give a simple proof that $e$ is transcendental over Liouville fields K$(\theta)$ where $\theta $ is a Liouville number with explicit very good rational approximations. The result extends to any Liouville field K$(\Theta )$ generated by a family $\Theta$ of Liouville numbers satisfying a Diophantine condition (the transcendence degree can be uncountable). This Diophantine condition is similar to the one appearing in Moser's theorem of simultanneous linearization of commuting holomorphic germs.

Keywords:

Small Divisors,
transcendental number theory,
Liouville Fields,
base of Neperian Logarithm,
simultaneous diophantine approximations.

Mathematics Subject Classification: 11J81, 11J82, 11J97, 11K60.

Citation:

Article Metrics

HTML views() PDF downloads(272) Cited by(0)

Diophantine conditions in small divisors and transcendental number theory

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Diophantine conditions in small divisors and transcendental number theory

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content