Additive averages of multiplicative correlation sequences and applications

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 t...

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Bibliographic Details
Published in:Journal d'analyse mathématique (Jerusalem) 2023-04, Vol.149 (2), p.719-761
Main Authors: Donoso, Sebastián, Le, Anh N., Moreira, Joel, Sun, Wenbo
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Analysis
Dynamical Systems and Ergodic Theory
Functional Analysis
Mathematics
Mathematics and Statistics
Number theory
Partial Differential Equations
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Summary: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.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-022-0264-x