On multiplicative recurrence along linear patterns

Abstract

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 $S=\{(an+b)/(cn+d):n\in \mathbb{N}\}$ 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 ${\mathrm{liminf}}_{n\to \infty }|f(an+b)-f(cn+d)|=0.$ 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 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 $(\mathbb{N},\times )$.

38 pages. Added a remark after Theorem 7.2, referee's comments incorporated. To appear in the Journal of the London Mathematical Society