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

We obtain asymptotics for sums of the form $$\sum_{n=1}^P e(\alpha_kn^k + \alpha_1n),$$ involving lower order main terms. As an application, we show that for almost all $\alpha_2 \in [0,1)$ one has $$\sup_{\alpha_1 \in [0,1)} \Big| \sum_{1 \le n \le P} e(\alpha_1(n^3+n) + \alpha_2 n^3) \Big| \ll P^{3/4 + \varepsilon},$$ and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schr\"odinger and Airy equations.