TY - JOUR
T1 - On generating functions in additive number theory, II
T2 - lower-order terms and applications to PDEs
AU - Brandes, J.
AU - Parsell, S. T.
AU - Poulias, C.
AU - Shakan, G.
AU - Vaughan, R. C.
N1 - Funding Information:
This paper emerged as part of a group discussion at the Heilbronn Focused Research Workshop on Decoupling and Efficient Congruencing, University of Bristol, 17th–21st June 2019. The authors are very grateful to the Heilbronn Institute for Mathematical Research, as well as the European Union’s Horizon 2020 research and innovation programme via grant agreement No. 695223, for funding this workshop, and to the University of Bristol for its hospitality. The work was further facilitated by a visit of three of the authors to Oberwolfach in November 2019. In addition, JB was supported by Starting Grant no. 2017-05110 of the Swedish Science Foundation (Vetenskapsrådet), CP was supported by a studentship sponsored by a European Research Council Advanced Grant under the European Union’s Horizon 2020 research and innovation programme via grant agreement No. 695223, and GS was supported by Simons Investigator Grant no. 376201 in the name of Ben Green.
Funding Information:
This paper emerged as part of a group discussion at the Heilbronn Focused Research Workshop on Decoupling and Efficient Congruencing, University of Bristol, 17th–21st June 2019. The authors are very grateful to the Heilbronn Institute for Mathematical Research, as well as the European Union’s Horizon 2020 research and innovation programme via grant agreement No. 695223, for funding this workshop, and to the University of Bristol for its hospitality. The work was further facilitated by a visit of three of the authors to Oberwolfach in November 2019. In addition, JB was supported by Starting Grant no. 2017-05110 of the Swedish Science Foundation (Vetenskapsrådet), CP was supported by a studentship sponsored by a European Research Council Advanced Grant under the European Union’s Horizon 2020 research and innovation programme via grant agreement No. 695223, and GS was supported by Simons Investigator Grant no. 376201 in the name of Ben Green.
Publisher Copyright:
© 2020, The Author(s).
PY - 2021/2
Y1 - 2021/2
N2 - We obtain asymptotics for sums of the form ∑n=1Pe(αknk+α1n),involving lower order main terms. As an application, we show that for almost all α2∈ [0 , 1) one has supα1∈[0,1)|∑1≤n≤Pe(α1(n3+n)+α2n3)|≪P3/4+ε,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.
AB - We obtain asymptotics for sums of the form ∑n=1Pe(αknk+α1n),involving lower order main terms. As an application, we show that for almost all α2∈ [0 , 1) one has supα1∈[0,1)|∑1≤n≤Pe(α1(n3+n)+α2n3)|≪P3/4+ε,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.
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U2 - 10.1007/s00208-020-02107-0
DO - 10.1007/s00208-020-02107-0
M3 - Article
C2 - 33603253
AN - SCOPUS:85099513742
SN - 0025-5831
VL - 379
SP - 347
EP - 376
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 1-2
ER -