On generating functions in additive number theory, II: lower-order terms and applications to PDEs

We obtain asymptotics for sums of the formSigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n),involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one hassup(alpha 1 is an element of[0,1)) | Sigma(1 \u3c= n \u3c= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| \u3c\u3c P3/4+epsilon,and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations