Fourier uniformity of bounded multiplicative functions in short intervals on average

  • Kaisa Matomäki
  • Maksym RadziwiłłEmail author
  • Terence Tao


Let \(\lambda \) denote the Liouville function. We show that as \(X \rightarrow \infty \),
$$\begin{aligned} \int _{X}^{2X} \sup _{\alpha } \left| \sum _{x < n \le x + H} \lambda (n) e(-\alpha n) \right| dx = o ( X H) \end{aligned}$$
for all \(H \ge X^{\theta }\) with \(\theta > 0\) fixed but arbitrarily small. Previously, this was only known for \(\theta > 5/8\). For smaller values of \(\theta \) 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 \(\lambda (n) \Lambda (n + h) \Lambda (n + 2h)\) over the ranges \(h < X^{\theta }\) and \(n < X\), and where \(\Lambda \) is the von Mangoldt function.



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.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Kaisa Matomäki
    • 1
  • Maksym Radziwiłł
    • 2
    Email author
  • Terence Tao
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  2. 2.Department of MathematicsCaltechPasadenaUSA
  3. 3.Department of MathematicsUCLALos AngelesUSA

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