November  2003, 9(6): 1401-1409. doi: 10.3934/dcds.2003.9.1401

Diophantine conditions in small divisors and transcendental number theory

1. 

Institut des Hautes Études Scientifiques, 35, Route de Chartres, 91440-Bures-Sur-Yvette, France

2. 

University of California Los Angeles, 520 Portola Plaza, Los Angeles, California 90095, United States

Received  October 2002 Revised  May 2003 Published  September 2003

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.
Citation: E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401
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