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Fourier uniformity of bounded multiplicative functions in short intervals on average

Matomäki, Kaisa and Radziwiłł, Maksym and Tao, Terence (2020) Fourier uniformity of bounded multiplicative functions in short intervals on average. Inventiones Mathematicae, 220 . pp. 1-58. ISSN 0020-9910. https://resolver.caltech.edu/CaltechAUTHORS:20190927-105826819

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Abstract

Let λ denote the Liouville function. We show that as X→∞, ∫^(2X)X supα∣∑x< n ≤ x+H λ(n)e(−αn)∣dx = o(XH) for all H ≥ X^θ with θ > 0 fixed but arbitrarily small. Previously, this was only known for θ > 5/8. For smaller values of θ this is the first “non-trivial” case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) 1-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of λ(n)Λ(n+h)Λ(n+2h) over the ranges h < X^θ and n < X, and where Λ is the von Mangoldt function.


Item Type:Article
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https://doi.org/10.1007/s00222-019-00926-wDOIArticle
https://rdcu.be/bR3ADPublisherFree ReadCube access
https://arxiv.org/abs/1812.01224arXivDiscussion Paper
https://doi.org/10.1007/s00222-019-00931-zDOICorrection
https://rdcu.be/b17FVPublisherFree ReadCube access -- Correction
Additional Information:© 2019 Springer-Verlag GmbH Germany, part of Springer Nature. Received: 21 January 2019; Accepted: 5 September 2019; Article First Online: 26 September 2019. KM was supported by Academy of Finland Grant No. 285894. MR was supported by an NSERC DG grant, the CRC program and a Sloan Fellowship. TT was supported by a Simons Investigator Grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF Grant DMS-1266164. Part of this paper was written while the authors were in residence at MSRI in Spring 2017, which is supported by NSF Grant DMS-1440140.
Errata:In the Acknowledgements, the second line should read: MR was supported by NSF grant DMS-1902063 and a Sloan Fellowship. Matomäki, K., Radziwiłł, M. & Tao, T. Correction to: Fourier uniformity of bounded multiplicative functions in short intervals on average. Invent. math. (2019). https://doi.org/10.1007/s00222-019-00931-z
Funders:
Funding AgencyGrant Number
Academy of Finland285894
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Canada Research Chairs ProgramUNSPECIFIED
Alfred P. Sloan FoundationUNSPECIFIED
Simons FoundationUNSPECIFIED
James and Carol Collins ChairUNSPECIFIED
Mathematical Analysis and Application Research Fund EndowmentUNSPECIFIED
NSFDMS-1266164
NSFDMS-1440140
NSFDMS-1902063
Record Number:CaltechAUTHORS:20190927-105826819
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20190927-105826819
Official Citation:Matomäki, K., Radziwiłł, M. & Tao, T. Fourier uniformity of bounded multiplicative functions in short intervals on average. Invent. math. 220, 1–58 (2020). https://doi.org/10.1007/s00222-019-00926-w
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:98905
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:27 Sep 2019 18:11
Last Modified:05 Mar 2020 18:08

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