Sarnak's Program for Erdős Sieves. Part II: Measure Systems and Applications

Abstract

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 $\hat{I}{\text{c◯}}_R$ (the set of $R-$admissible sets) and show how they are related. We show that the system associated to $\hat{I}{\text{c◯}}_R$ 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.

56 pages