Additive averages of multiplicative correlation sequences and applications

Abstract

We study sets of recurrence, in both measurable and topological settings, for actions of $(\mathbb{N},\times )$ and $({\mathbb{Q}}^{>0},\times )$. 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 $\{(an+b{)}^{\ell }/(cn+d{)}^{\ell }:n\in \mathbb{N}\}$ 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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• Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer
• Sarnak's Conjecture for nilsequences on arbitrary number fields and applications