  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.
