Discrete Fourier restriction associated with KdV equations

In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear term $ F(u)\p_x u$ provided $F\in C^5$ and the initial data $\phi\in H^s$ with $s>1/2$.