Sarnak's Program for Erd\H{o}s Sieves. Part II: Measure Systems and Applications

TL;DR

This paper extends Sarnak's program to Erd ext{"o}s sieves in algebraic number fields, analyzing associated measure systems and applying results to sumsets, polynomial squarefree values, and prime number theorems.

Contribution

It generalizes Sarnak's measure-theoretic approach to Erd ext{"o}s sieves over algebraic integers and establishes connections with ergodic rotations and spectral analysis.

Findings

The system for $oldsymbol{ ext{Ω}_R}$ is isomorphic to an ergodic rotation of a compact abelian group.

Computed the spectrum of the associated dynamical system.

Derived results on sumsets, polynomial squarefree values, and prime number distribution.

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 \'etale $\mathbb{Q}-$algebra $K$, which we denote by Erd\H{o}s 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 $\Omega_R$ (the set of $R-$admissible sets) and show how they are related. We show that the system associated to $\Omega_R$ 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.