Non recursive proof of the KAM theorem

: A selfcontained proof of the KAM theorem in the Thirring model is discussed, completely relaxing the "strong diophantine property" hypothesis used in previous papers. Keywords: KAM, invariant tori, classical mechanics, perturbation theory, chaos 1. Introduction In [G] a selfcontained proof of the KAM theorem in the Thirring model is discussed, under the hypothesis that the rotation vectors ~! 0 verify a strong diophantine property. At the end of the same paper a heuristic argument is given to show that in fact such a hypothesis can be relaxed. In the present work we develop the heuristic argument into an extension of the KAM theorem proof described in [G]; the extension applies to rotations vectors verifying only the usual diophantine condition. It is a proof again based on Eliasson's method, [E]. In our opinion this shows that a hypothesis like the strong diophantine property of [G], or something similar to it, is very natural as it simplifies the structure of the proof by separati..