Explicit estimates on the measure of primary KAM tori

From KAM Theory it follows that the measure of phase points which do not lie on Diophantine, Lagrangian, “primary” tori in a nearly–integrable, real–analytic Hamiltonian system is $O(\sqrt{\epsilon })$, if $\epsilon$ is the size of the perturbation. In this paper we discuss how the constant in front of $\sqrt{\epsilon }$ depends on the unperturbed system and in particular on the phase–space domain.