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Fourier uniformity of bounded multiplicative functions in short intervals on average

  • Kaisa Matomäki
  • Maksym RadziwiłłEmail author
  • Terence Tao
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Abstract

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.
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Notes

Acknowledgements

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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