On multiplicative recurrence along linear patterns

In a recent article, Donoso, Le, Moreira and Sun studied sets of recurrence for actions of the multiplicative semigroup $(\mathbb{N},\times )$ and provided some sufficient conditions for sets of the form $S=\{(an+b)\mathrm{/}(cn+d)\text{\hspace{0.17em}⁣}:n\in \mathbb{N}\}$ to be sets of recurrence for such actions. A necessary condition for $S$ to be a set of multiplicative recurrence is that for every completely multiplicative function $f$ taking values on the unit circle, we have that ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}\mathrm{\mid }f(an+b)-f(cn+d)\mathrm{\mid }=0.$ In this article, we fully characterize the integer quadruples $(a,b,c,d)$ which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel concerning the pair $(n,n+1)$, as well as some results of Donoso, Le, Moreira and Sun. In addition, we prove that, under the same conditions on $(a,b,c,d)$, the set $S$ is a set of recurrence for finitely generated actions of $(\mathbb{N},\times )$.