Erdős筛法的Sarnak计划。第二部分：度量系统与应用
Sarnak's Program for Erdős Sieves. Part II: Measure Systems and Applications
AI总结：
本文扩展了Sarnak的计划，研究Erdős筛法的度量系统与应用，包括与ergodic旋转的同构性、无限和集结果及素数定理。
摘要：
本文是两部分文章的第二部分，我们扩展了Sarnak的计划，使其适用于移除某些无限理想集$\mathcal{B}$的同余类的集合，我们称之为Erdős筛法。给定一个筛子$R$，我们定义集合$\mathcal{F}_R$为不在$R$的同余类中的代数整数。我们为每个筛子关联两个测度理论动力系统$X_R$（$\mathcal{F}_R$的轨道闭包）和$Ω_R$（$R$-允许的集合），并展示它们之间的关系。我们证明与$Ω_R$相关的系统与紧致交换群的ergodic旋转同构，并计算其谱。作为应用，我们展示了整数中的无限和集结果，研究了$\mathcal{F}_R$为某些多项式的无平方因子值的情况，并展示了$R$-自由数的素数定理。
英文摘要：
This paper is the second part of a two-part article where we generalize Sarnak's program to sets where we remove congruence classes modulo some infinite set $\mathcal{B}$ of ideals of an étale $\mathbb{Q}-$algebra $K$, which we denote by Erdős sieves. Given a sieve $R$ we define the set $\mathcal{F}_R$ of algebraic integers in $K$ not contained in any of the congruence classes of $R$. We associate to each sieve two measure-theoretical dynamical systems $X_R$ (the orbit closure of $\mathcal{F}_R$) and $Ω_R$ (the set of $R-$admissible sets) and show how they are related. We show that the system associated to $Ω_R$ is isomorphic to an ergodic rotation of a compact abelian group, and compute its spectrum. As applications we show results about infinite sumsets in the integers, investigate the case where $\mathcal{F}_R$ is the squarefree values of some polynomial, and show a prime number theorem for $R-$free numbers.
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