Sarnak's Program for Erd\H{o}s Sieves. Part I: Topological Dynamics and Light Tails

Abstract

This paper is the first 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. We define some light tail conditions on a sieve $R$, and show how these are related to the genericity under the Mirsky measure of the set of $R-$free numbers, which are the algebraic integers of $K$ not contained in any of the congruence classes in $R$. We also show that Erd\H{o}s $\mathcal{B}-$free systems in any \'etale $\mathbb{Q}-$algebra satisfy these light tail conditions, so our results generalize Sarnak's program to Erd\H{o}s $\mathcal{B}-$free systems over any \'etale $\mathbb{Q}-$algebra.