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 \begin{document}$ \omega $\end{document} of an integrable system by a trigonometric polynomial of degree \begin{document}$ N $\end{document} perturbation \begin{document}$ R_N $\end{document} with \begin{document}$ \|R_N\|_{C^r}<\epsilon $\end{document} . We obtain a relation among \begin{document}$ N $\end{document} , \begin{document}$ r $\end{document} , \begin{document}$ \epsilon $\end{document} and the arithmetic property of \begin{document}$ \omega $\end{document} , for which the area-preserving map admit no invariant circles with \begin{document}$ \omega $\end{document} .