We review V. I. Arnold’s 1963 celebrated paper  Proof of A. N. Kolmogorov’s Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold’s scheme, one can get “sharp” asymptotic quantitative conditions (as ε → 0, ε being the strength of the perturbation). All constants involved are explicitly computed.
Chierchia, L., & Koudjinan, C.E. (2019). V. I. Arnold’s “Pointwise” KAM Theorem. REGULAR & CHAOTIC DYNAMICS, 24(6), 583-606 [10.1134/S1560354719060017].