MOSER’S TWIST THEOREM REVISITED

The paper’s Theorem 1 proves that for Gn(x,x′)=½(x−x′)^2+q_n^{-(4+ε)}V(q_nx′), the exact twist map fn admits, for each constant‑type α and all large n, an invariant circle with rotation number α that is a C^{2+ε′} graph (any ε′<ε) . The candidate solution derives the same result by showing fn→f̄ in C^{3+ε′} and invoking Herman’s finite‑smoothness KAM theorem for constant‑type α; this aligns with the paper’s observation that ‖fn−f̄‖_{C^{3+ε′}}→0 as n→∞ and that the circle is a graph . The proofs, however, differ: the paper uses a Katznelson–Ornstein/Mather–type variational approach, not KAM .