invariant tori) which is new, and the condition  is only required for this part of the statement. Under this condition, we will see in the proof of Theorem \[Th2\] that for any fixed $\tau>n-1$, for any $\gamma>0$ and for $\varepsilon$ sufficiently small, the tori in the set of KAM invariant tori $$\label{def.tOmega} \mathcal K'=\bigcup_{\omega_{*} \in \Omega_{\gamma,\tau}'}\mathcal{T}_{\omega_{*}}^\varepsilon$$ are doubly exponentially stable with exponent $\frac{1}{{\alpha}(\tau+1)}$. Since $\tau$ can be chosen to be any number strictly smaller than $n-1$, we can reach any exponent $u$ with $u<\frac{1}{{\alpha}n}$. Here, we restricted, following [@Pop04], the frequencies to the subset $\Omega_{\gamma,\tau}'$ of vectors in $\Omega_{\gamma,\tau}$ which have positive Lebesgue density in $\Omega_{\gamma,\tau}$; that is $\omega \in \Omega_{\gamma,\tau}'$ if for any neighborhood $O$ of $\omega$ in $\Omega$, the Lebesgue measure of $O \cap \Omega_{\gamma,\tau}$ is positive. No measure is lost due to this restriction since the sets $\Omega_{\gamma,\tau}$ and $\Omega_{\gamma,\tau}'$ have the same Lebesgue measure.

For every fixed couple $(\gamma,\tau)$, the positive constants $\varepsilon_*$, $r_*$ and $C$, characterizing the smallness of the perturbation as well as the constants that appear in the double exponential stability of the tori in $\mathcal K'$ of , depend only on $n$, $\gamma$, $\tau$, $\alpha$, $L$, $||h||_{\alpha,L,\bar{D}}$, $||f||_{\alpha,L,\bar{D}}$ and on some uniform 