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Sarnak's Program for Erdős Sieves. Part I: Topological Dynamics and Light Tails

Francisco Araújo

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 étale $\mathbb{Q}-$algebra $K$, which we denote by Erdős 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 Erdős $\mathcal{B}-$free systems in any étale $\mathbb{Q}-$algebra satisfy these light tail conditions, so our results generalize Sarnak's program to Erdős $\mathcal{B}-$free systems over any étale $\mathbb{Q}-$algebra.

Sarnak's Program for Erdős 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 of ideals of an étale algebra , which we denote by Erdős sieves. We define some light tail conditions on a sieve , and show how these are related to the genericity under the Mirsky measure of the set of free numbers, which are the algebraic integers of not contained in any of the congruence classes in . We also show that Erdős free systems in any étale algebra satisfy these light tail conditions, so our results generalize Sarnak's program to Erdős free systems over any étale algebra.
Paper Structure (12 sections, 43 theorems, 247 equations, 1 table)

This paper contains 12 sections, 43 theorems, 247 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices in étale $\mathbb{Q}-$algebras
  4. Dynamical Systems
  5. Sieves and $R-$free numbers
  6. Shift Spaces
  7. Density of $R-$free numbers
  8. Topological Dynamical Systems
  9. Sieves with Light Tails
  10. Repeated Patterns in $\mathcal{F}_R$
  11. Sieves with Strong Light Tails
  12. Weak and Strong Light Tails

Key Result

Theorem 1.1

233434 Let $R$ be an Erdős sieve. For a given Følner sequence $I_N$, the following are equivalent.

Theorems & Definitions (117)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 107 more