Document Type
Article
Publication Date
12-2020
Abstract
We obtain asymptotics for sums of the form
Sigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n),
involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one has
sup(alpha 1 is an element of[0,1)) | Sigma(1 <= n <= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| << P3/4+epsilon,
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.
Publication Title
Mathematische Annalen
ISSN
1432-1807
Publisher
Springer
DOI
10.1007/s00208-020-02107-0
Recommended Citation
Brandes, J., Parsell, S. T., Poulias, C., Shakan, G., & Vaughn, R. C. (2020). On generating functions in additive number theory, II: lower-order terms and applications to PDEs. Mathematische Annalen http://dx.doi.org/10.1007/s00208-020-02107-0