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.

中文翻译：

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关于加性数论中的生成函数，II：低阶项及其在偏微分方程中的应用

我们获得 $$ \sum_{n=1}^P e(\alpha_kn^k + \alpha_1n) 形式的和的渐近性，$$ 涉及低阶主项。作为一个应用，我们证明几乎所有的 $\alpha_2 \in [0,1)$ 都有 $$ \sup_{\alpha_1 \in [0,1)} \Big| \sum_{1 \le n \le P} e(\alpha_1(n^3+n) + \alpha_2 n^3) \Big| \ll P^{3/4 + \varepsilon}, $$ 并且在适当的意义上这是最好的。这使我们能够改进薛定谔方程和艾里方程解的分形维数的界限。