It is known that a Diophantine quasi-periodic torus with frequency \(\omega \in \Omega _{\tau }^d\) of a \(C^{l}\) Hamiltonian is effectively stable for a time T(r) that is polynomial on the inverse of the distance to the torus, that we denote by r, with exponent \(1+(l-2)/(\tau +1)\). It is also known that a Diophantine quasi-periodic torus of a Gevrey Hamiltonian \(H\in G^{\alpha ,L}\) is effectively stable for an exponentially long time on the inverse of the distance to the torus with exponent \(1/(\alpha (1+\tau ))\). In this note, we see that following the methods in [11] one can show the almost optimality of these exponents. We also show that, for a dense subset of non-resonant vectors, for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians, the naive lower bound \(T(r)\ge Cr^{-1}\) is optimal in terms of the exponent.

已知丢番图准周期环面的频率为\(\omega \in \Omega _{\tau }^d\) \(C^{l}\) 哈密顿量在一段时间内有效稳定 T) 是到环面的距离的倒数多项式，我们用 r(，指数为 \(1+(l-2)/(\tau +1)\)。还已知 Gevrey Hamiltonian 的丢番图准周期环面 \(H\in G^{\alpha ,L}\) 实际上是在到环面的距离的倒数上保持指数长时间稳定，指数为 \(1/(\alpha (1+\tau ))\)。在这篇文章中，我们看到遵循[11]中的方法可以显示这些指数的几乎最优性。我们还表明，对于非共振向量的稠密子集，对于有限微分和 Gevrey Hamiltonian 的准周期环面，朴素下界 \(T(r)\ge Cr^ {-1}\) 就指数而言是最佳的。