Diophantine conditions in small divisors and transcendental number theory

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.