标题： 关于线性模式上的乘法递推 显示英文标题

标题： On multiplicative recurrence along linear patterns

摘要： 在最近的一篇文章中，Donoso、Le、Moreira 和 Sun 研究了乘法半群$(\mathbb{N}, \times)$的作用下的回归集，并提供了形式为$S=\{(an+b)/(cn+d) \colon n \in \mathbb{N} \}$的集合成为此类作用下回归集的一些充分条件。 $S$成为乘法回归集的必要条件是，对于每个取值于单位圆上的完全乘法函数$f$，我们有 $\liminf_{n \to \infty} |f(an+b)-f(cn+d)|=0.$。在这篇文章中，我们完全描述了满足后一种性质的整数四元组$(a,b,c,d)$。 我们的结果推广了 Klurman 和 Mangerel 关于对$(n,n+1)$的结果，以及 Donoso、Le、Moreira 和 Sun 的一些结果。 此外，我们证明，在对$(a,b,c,d)$的相同条件下，集合$S$是$(\mathbb{N}, \times)$的有限生成作用的回归集。 显示英文摘要

摘要： 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 $(\mathbb{N}, \times)$.