On generating functions in additive number theory, II: lower-order terms and applications to PDEs
Journal article, 2021
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.
Author
Julia Brandes
Chalmers, Mathematical Sciences, Algebra and geometry
University of Gothenburg
S. T. Parsell
West Chester University
C. Poulias
University of Bristol
G. Shakan
University of Oxford
R. C. Vaughan
Pennsylvania State University
Mathematische Annalen
0025-5831 (ISSN) 1432-1807 (eISSN)
Vol. 379 1-2 347-376Higher-dimensional structures on hypersurfaces
Swedish Research Council (VR) (2017-05110), 2018-01-01 -- 2021-12-01.
Subject Categories
Computational Mathematics
Discrete Mathematics
Mathematical Analysis
DOI
10.1007/s00208-020-02107-0