On multiplicative recurrence along linear patterns

Abstract

Donoso, Le, Moreira, and Sun (J. Anal. Math. 149 (2023), 719–761) study sets of recurrence for actions of the multiplicative semigroup $(N,\times )$ and provide some sufficient conditions for sets of the form $S=\left\{\right(an+b)/(cn+d):n\in N\}$ 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 ${lim\; inf}_{n\to \infty }|f(an+b)-f(cn+d)|=0$. 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 (Math. Ann. 372 (2018), 651–697) concerning the pair $(n,n+1)$, as well as some results from (Donoso, Le, Moreira, and Sun, J. Anal. Math. 149 (2023), 719–761). 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 $(N,\times )$.