V.I. Arnold’s “Global” KAM Theorem and geometric measure estimates

The paper proves a global, Whitney-smooth KAM theorem for H(y,x)=K(y)+εP with explicit constants and the key iso-frequency identity By* K* ∘ Y* = By K, together with the sharp logarithmic loss ℓ = 8(s−s*)^{-1} log(ε̂^{-1}) and the quantitative bounds on Y*, u*, v*, and B_x u* (Theorem 1 and its proof in Appendix A) . The candidate solution reproduces the same parameter choices (ε̂ := (M ε P)/α^2, θ := ML), the iso-frequency step (By1 K1 ∘ G = By K), the quadratic recurrence, the Fourier cutoff K_n ≍ 4 δ_n^{-1} log(ε_n^{-1}), and the Whitney-smooth convergence on a nowhere dense action Cantor set, yielding exactly the estimates stated in the paper. Aside from a minor overstatement that the limiting action set is “perfect” (the paper only guarantees nowhere dense, and notes that closed Diophantine sets may have isolated points), the approaches and conclusions match closely, using substantially the same Arnold-type KAM scheme and analytic tools .