We obtain asymptotics for sums of the form
Sigma(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 has
sup(alpha 1 is an element of[0,1)) | Sigma(1 <= n <= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| << 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.
Brandes, J., Parsell, S. T., Poulias, C., Shakan, G., & Vaughn, R. C. (2020). On generating functions in additive number theory, II: lower-order terms and applications to PDEs. Mathematische Annalen http://dx.doi.org/10.1007/s00208-020-02107-0