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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 almostall $\alpha_2 \in [0,1)$ 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 improvebounds for the fractal dimension of solutions to the Schr\"odinger and Airyequations.

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