Quantitative destruction of invariant circles

For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency $\omega$ of an integrable system by a trigonometric polynomial of degree $N$ perturbation $R_N$ with $\|R_N\|_{C^r}<\epsilon$. We obtain a relation among $N$, $r$, $\epsilon$ and the arithmetic property of $\omega$, for which the area-preserving map admit no invariant circles with $\omega$.