On multiplicative recurrence along linear patterns

TL;DR

This paper characterizes when certain rational linear patterns are recurrence sets for multiplicative actions, extending previous results and establishing conditions under which these sets exhibit recurrence properties.

Contribution

It fully characterizes integer quadruples for which the associated sets are recurrence sets, generalizing prior results and applying to finitely generated multiplicative actions.

Findings

Complete characterization of quadruples (a,b,c,d) satisfying the recurrence condition.

Extension of previous results on the pair (n,n+1) to broader linear patterns.

Proof that these sets are recurrence sets for finitely generated actions.

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