  It was conjectured by Herman that an analytic Lagrangian Diophantine
quasi-periodic torus $\mathcal{T}_0$, invariant by a real-analytic Hamiltonian
system, is always accumulated by a set of positive Lebesgue measure of other
Lagrangian Diophantine quasi-periodic invariant tori. While the conjecture is
still open, we will prove the following weaker statement: there exists an open
set of positive measure (in fact, the relative measure of the complement is
exponentially small) around $\mathcal{T}_0$ such that the motion of all initial
conditions in this set is "effectively" quasi-periodic in the sense that they
are close to being quasi-periodic for an interval of time which is doubly
exponentially long with respect to the inverse of the distance to
$\mathcal{T}_0$. This open set can be thought as a neighborhood of a
hypothetical invariant set of Lagrangian Diophantine quasi-periodic tori, which
may or may not exist.
