  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.
