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

J. Brandes; S. T. Parsell; C. Poulias; G. Shakan; R. C. Vaughan

doi:10.1007/s00208-020-02107-0

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

J. Brandes
, S. T. Parsell
, C. Poulias
, G. Shakan
, R. C. Vaughan

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

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.

Original language	English (US)
Pages (from-to)	347-376
Number of pages	30
Journal	Mathematische Annalen
Volume	379
Issue number	1-2
DOIs	https://doi.org/10.1007/s00208-020-02107-0
State	Published - Feb 2021

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s00208-020-02107-0

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title = "On generating functions in additive number theory, II: lower-order terms and applications to PDEs",

abstract = "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 Schr{\"o}dinger and Airy equations.",

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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.

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.

UR - https://www.scopus.com/pages/publications/85099513742

UR - https://www.scopus.com/pages/publications/85099513742#tab=citedBy

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 -

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

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