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Metric results on summatory arithmetic functions on Beatty sets

Volume 197 / 2021

Marc Technau, Agamemnon Zafeiropoulos Acta Arithmetica 197 (2021), 93-104 MSC: Primary 11B83; 11J83, 11K65. DOI: 10.4064/aa200128-10-6 Published online: 31 August 2020

Abstract

Let $f\colon \mathbb N \rightarrow \mathbb C $ be an arithmetic function and consider the Beatty set $\mathcal{B} (\alpha ) = \{ \lfloor {n\alpha }\rfloor : n\in \mathbb N \}$ associated to a real number $\alpha $, where $\lfloor {\xi }\rfloor$ denotes the integer part of a real number $\xi $. We show that the asymptotic formula \[\biggl| \sum _{\substack { 1\leq m\leq x \\ m\in \mathcal{B} (\alpha ) }} f(m) - \frac {1}{\alpha } \sum _{1\leq m\leq x} f(m) \biggr|^2 \ll _{f,\alpha ,\varepsilon } (\log x) (\log \log x)^{3+\varepsilon } \sum _{1\leq m\leq x} | {f(m)}|^2 \] holds for almost all $\alpha \gt 1$ with respect to the Lebesgue measure. This significantly improves an earlier result due to Abercrombie, Banks, and Shparlinski. The proof uses a recent Fourier-analytic result of Lewko and Radziwiłł based on the classical Carleson–Hunt inequality.

Moreover, using a probabilistic argument, we establish the existence of functions $f\colon \mathbb N \to \{\pm 1\}$ for which the above error term is optimal up to logarithmic factors.

Authors

  • Marc TechnauInstitut für Analysis und Zahlentheorie
    TU Graz
    Kopernikusgasse 24/II
    8010 Graz, Austria
    e-mail
  • Agamemnon ZafeiropoulosDepartment of Mathematical Sciences
    Norwegian University of Science and Technology
    NO-7491 Trondheim, Norway
    e-mail

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