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

We obtain asymptotics for sums of the form $$\sum _{n=1}^{P}e({\alpha }_k{n}^k+{\alpha }_{1}n),$$ involving lower order main terms. As an application, we show that for almost all ${\alpha }_2\in [0,1)$ one has $$\underset{{\alpha }_1\in [0,1)}{\sup }\mid \sum _{1\le n\le P}e({\alpha }_1(n^3+n)+{\alpha }_2{n}^3)\mid \ll P^{3\mathrm{/}4+\epsilon },$$ and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schr"odinger and Airy equations.