DCDS
Diophantine conditions in small divisors and transcendental number theory
E. Muñoz Garcia R. Pérez-Marco
Discrete & Continuous Dynamical Systems - A 2003, 9(6): 1401-1409 doi: 10.3934/dcds.2003.9.1401
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 Liouville Fields transcendental number theory base of Neperian Logarithm simultaneous diophantine approximations.

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