with respect to $m \sim r^{-a}$. In the case of an elliptic fixed point, these estimates have been proved but the calculations do not apply directly to the case of invariant tori. However a more general result (valid for Gevrey smooth systems) in the case of an invariant torus has been obtained by Mitev and Popov in [@MP09]. Indeed, by Borel’s theorem the formal Birkhoff series $h_{\infty}$ can be realized as the Taylor series at $I=0$ of some smooth function $H_*(I)$; it is proved in [@MP09] that one can choose such a function $H_*$ in a Gevrey class, and this gives immediately the desired growth on the coefficients of $h_{\infty}$. More precisely, if $H$ is $\alpha$-Gevrey and $\omega$ satisfies , then setting $\beta=\alpha(1+\tau)+1$, one can find a symplectic transformation $\Phi$ which is $(\alpha,\beta)$-Gevrey, such that $H \circ \Phi(\theta,I)=H_*(I)+R_*(\theta,I)$, where $H_*$ is $\beta$-Gevrey, $R_*$ is $(\alpha,\beta)$-Gevrey and flat at $I=0$. It follows that on a $r$-neighborhood of $I=0$, $R_*$ is exponentially small with respect to $r^{-(\beta-1)^{-1}}=r^{-\alpha^{-1}a}$. Now, if in addition, the Birkhoff polynomial $H_{m_0}$ is assumed to be stably steep, one can prove as in [@BFN15] that any sufficiently smooth function, whose Taylor expansion at some point coincides with $H_{m_0}$, is 