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

We obtain asymptotics for sums of the form ∑ n = 1 P e α k n k + α 1 n , involving lower order main terms. As an application, we show that for almost all α 2 ∈ [ 0 , 1 ) one has sup α 1 ∈ [ 0 , 1 ) | ∑ 1 ≤ n ≤ P e α 1 n 3 + n + α 2 n 3 | ≪ P 3 / 4 + ε , and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations.