Diophantine conditions in small divisors and transcendental number theory

E. Muñoz Garcia; R. Pérez-Marco

doi:10.3934/dcds.2003.9.1401

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

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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: October 2002

Revised: May 2003

Published: November 2003

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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.

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