Moser's twist theorem revisited

Inspired by the work of Katznelson and Ornstein, we present a short way to achieve the almost optimal regularity in Moser's twist theorem. Specifically, for an integrable area-preserving twist map, the invariant circle with a given constant type frequency $\alpha$ persists under a small perturbation (dependent on $\alpha$) of class $C^{3+\epsilon}$. This result was initially established independently by Herman and R\"{u}ssmann in 1983. Our method differs essentially from their approaches.