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 Schrodinger and Airy equations.
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