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Fourier uniformity of bounded multiplicative functions in short intervals on average

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A Correction to this article was published on 20 November 2019

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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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  • 20 November 2019

    The original version of this article unfortunately contains a mistake.

Notes

  1. All the results for \(\lambda \) discussed here are also applicable to the Möbius function \(\mu \) with only minor changes to the arguments; we leave the details to the interested reader.

  2. Our conventions for asymptotic notation are given at the end of this introduction.

  3. By applying Hölder’s inequality to (3), it is also possible to obtain an averaged version of (2) over all shifts \(h_1, \ldots , h_k\); see [23] for details.

  4. The role of the parameter X here is mostly to control the size of t. It is not important that the sum over p runs up to X; it could run up to \(X^B\) for any \(B > 0\), since primes in \((X^\alpha , X^\beta ]\) contribute only \(O_{\alpha , \beta }(1)\) to the distance.

  5. More precisely, \(\frac{J_1}{q_1}\) and \(\frac{J_2}{q_2}\) will both intersect a third interval \(\frac{I}{p}\), but this is almost the same as requiring that these intervals intersect each other, as they are all of comparable size; see Fig. 4. For sake of this discussion, we ignore this technical distinction.

  6. It should also be possible to work in a purely continuous setting, replacing various summations in our arguments with appropriately normalized integrals, using Fubini’s theorem in place of double counting arguments, allowing the intervals under consideration to overlap each other, and with various graph-theoretic inequalities replaced by their continuous counterparts. We leave the details of this alternate arrangement of the argument to the interested reader.

  7. At the cost of worsening the dependence on \(\eta \) slightly, one could also use the standard large sieve inequality [26] here, combined with Lemma 2.4 below.

  8. If one were to extend the arguments here to smaller values of H, one would need to pay more attention as to the precise dependence of these constants on k.

  9. This bound also follows from the work of Sidorenko [29], as the graph consisting of two k-cycles (with k even) connected by an edge is one of the confirmed cases of Sidorenko’s conjecture.

  10. The \(\frac{1}{100} 1_{a_1 \ne a_2}\) term is present only to keep the metric Z from degenerating, but otherwise plays no role in the argument; if one prefers, one could drop this term and observe that Corollary 5.2 also applies to degenerate metric spaces.

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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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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Turku, 20014, Turku, Finland

    Kaisa Matomäki

  2. Department of Mathematics, Caltech, 1200 E California Blvd, Pasadena, CA, 91125, USA

    Maksym Radziwiłł

  3. Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA, 90095, USA

    Terence Tao

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  1. Kaisa Matomäki
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Correspondence to Maksym Radziwiłł.

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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

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