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arXiv · 2001.05629

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

Julia Brandes·Scott T. Parsell·Konstantinos Poulias·George Shakan·Robert C. Vaughan
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Abstract

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 Schr\"odinger and Airy equations.

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Julia Brandes, Scott T. Parsell, Konstantinos Poulias, George Shakan, Robert C. Vaughan. 2020-01-16. On generating functions in additive number theory, II: Lower-order terms and applications to PDEs. https://arxiv.org/abs/2001.05629

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