A Diophantine duality applied to the KAM and Nekhoroshev theorems

TL;DR

This paper introduces a novel geometric approach to Diophantine approximation that simplifies the proof of stability theorems like KAM and Nekhoroshev for quasi-periodic solutions, avoiding traditional small divisor issues.

Contribution

It develops a new method based on geometry of numbers to relate dual Diophantine problems, leading to simplified proofs of stability results in perturbation theory.

Findings

Established a Nekhoroshev-type stability result with a partial normal form.

Constructed an inverted normal form under Bruno-Rüssmann condition, recovering classical KAM theorem.

Avoided classical small divisors estimates using geometric Diophantine duality.

Abstract

In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a consequence, we develop a new approach to the perturbation theory for quasi-periodic solutions dealing only with periodic approximations and avoiding classical small divisors estimates. We obtain two results of stability, in the spirit of the KAM and Nekhoroshev theorems, in the model case of a perturbation of a constant vector field on the $n$-dimensional torus. Our first result, which is a Nekhoroshev type theorem, is the construction of a "partial" normal form, that is a normal form with a small remainder whose size depends on the Diophantine properties of the vector. Then, assuming our vector satisfies the Bruno-R\"ussmann condition, we construct an "inverted" normal form, recovering the classical KAM theorem of Kolmogorov, Arnold and Moser for constant vector fields on torus.