论文标题
在添加数理论中生成函数，ii：较低阶段的术语和应用PDES
On generating functions in additive number theory, II: Lower-order terms and applications to PDEs
论文作者
论文摘要
我们获得了形式和的渐近学 $$ \ sum_ {n = 1}^p e（α_kn^k +α_1n）， $$ 涉及较低的主术语。作为一个应用程序，我们表明几乎所有$α_2\ in [0,1）$ $$ \ sup_ {α_1\ in [0,1）} \ big | \ sum_ {1 \ le n \ le p} e（α_1（n^3 + n） +α_2n^3）\ big | \ ll p^{3/4 + \ varepsilon}，$$ 从适当的意义上讲，这是最好的。这使我们能够改善溶液对Schrödinger和通风方程的分形维度的界限。
We obtain asymptotics for sums of the form $$ \sum_{n=1}^P e(α_kn^k + α_1n), $$ involving lower order main terms. As an application, we show that for almost all $α_2 \in [0,1)$ one has $$ \sup_{α_1 \in [0,1)} \Big| \sum_{1 \le n \le P} e(α_1(n^3+n) + α_2 n^3) \Big| \ll P^{3/4 + \varepsilon}, $$ and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations.
