Discrete Fourier restriction associated with Schrodinger equations

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgains level set result on Strichartz estimates associated with Schrodinger equations on torus. Some sharp estimates on $L^{frac{2(d+2)}{d}}$ norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on $L^2(mathbb Z)$.