Additive averages of multiplicative correlation sequences and applications

TL;DR

This paper investigates recurrence properties in multiplicative systems, demonstrating that autocorrelation sequences have positive averages and establishing criteria for multiplicative recurrence in specific sets, with applications to number theory.

Contribution

It introduces new criteria for multiplicative recurrence and connects autocorrelation sequences with positive averages in multiplicative systems.

Findings

Autocorrelation sequences of positive functions have positive additive averages.

Criteria established for sets of the form ((an+b)^{}/(cn+d)^{}) to be multiplicative recurrence.

Reveals implications for number theory, including results on multiplicative functions and the Omega function.

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.