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Additive averages of multiplicative correlation sequences and applications

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Abstract

We study sets of recurrence, in both measurable and topological settings, for actions of (ℕ, ×) and (ℚ>0, ×). In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form {(an+b)1/(cn+d)ℓ: n ∈ ℕ} are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.

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Acknowledgements

We thank Florian Richter for help with the proof of Lemma 3.5 and thank Vitaly Bergelson for helpful conversations. We also thank the referee for suggestions that improve the organization and readability of the paper.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático CNRS IRL 2807, Universidad de Chile, Beauchef 851, Santiago, Chile

    Sebastián Donoso

  2. Department of Mathematics, Ohio State University, 231 W. 18th Ave., Columbus, OH, 43210, USA

    Anh N. Le

  3. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Joel Moreira

  4. Department of Mathematics, Virginia Polytechnic Institute and State University, 225 Stanger Street, Blacksburg, VA, 24061-1026, USA

    Wenbo Sun

Authors
  1. Sebastián Donoso
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  2. Anh N. Le
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  3. Joel Moreira
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  4. Wenbo Sun
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Corresponding author

Correspondence to Joel Moreira.

Additional information

The first author was supported by ANID/Fondecyt/1200897 and Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for centers of excellence from ANID-Chile.

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Donoso, S., Le, A.N., Moreira, J. et al. Additive averages of multiplicative correlation sequences and applications. JAMA 149, 719–761 (2023). https://doi.org/10.1007/s11854-022-0264-x

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  • DOI: https://doi.org/10.1007/s11854-022-0264-x

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