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Published April 2020 | Submitted
Journal Article Open

Fourier uniformity of bounded multiplicative functions in short intervals on average


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.

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.


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

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