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

This paper generalizes Sarnak's program to Erdős sieves over étale $\mathbb{Q}$ -algebras by establishing light tail conditions that ensure the genericity of $R$ -free numbers under the Mirsky measure, thereby extending the framework to any such algebraic system.

Original authors: Francisco Araújo

Published 2026-03-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Francisco Araújo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The Great Cosmic Filter

Imagine you have a giant, infinite bucket of sand. Each grain of sand represents a number (or an algebraic integer). Now, imagine you want to sift through this bucket to find only the "special" grains.

In mathematics, this process is called sieving. You take a sieve (a colander) with holes of different sizes. If a grain of sand is too big to fit through a hole, it gets caught. If it fits, it falls through.

For over a century, mathematicians have studied a very specific type of sieve: The Squarefree Sieve.

The Analogy: Imagine you have a sieve that catches any number divisible by a perfect square (like 4, 9, 16, 25).
The Result: The grains that fall through are the "squarefree numbers" (numbers like 1, 2, 3, 5, 6, 7, 10... but not 4 or 8).
The Mystery: Mathematicians wanted to know: If I shake this sieve randomly, do the grains that fall through behave in a predictable, "random" way?

This is where Sarnak's Program comes in. It's a famous set of rules proposed by mathematician Peter Sarnak. He predicted that these "falling grains" (squarefree numbers) have a very specific, beautiful structure. They are "generic," meaning they represent the average behavior of the whole system.

The Problem: What if the Sieve is Weird?

The old rules only worked for the standard squarefree sieve. But what if you change the rules?

What if you remove numbers that are divisible by $p^2$ plus some other random numbers?
What if you are working not just with normal numbers, but with complex "algebraic" numbers (like points on a grid in higher dimensions)?

This is where Erdős Sieves come in. Named after the legendary mathematician Paul Erdős, these are sieves where you remove specific "congruence classes" (patterns of numbers) modulo an infinite list of ideals.

The Paper's Goal:
Francisco Araújo wants to prove that Sarnak's beautiful rules still work even when the sieve is weird, complex, and operates in these high-dimensional algebraic worlds.

The Key Concept: "Light Tails"

To make this work, the author introduces a new concept called "Light Tails."

The Analogy of the Heavy Tail:
Imagine you are fishing with a net.

Strong Light Tails: You cast the net, and almost all the fish you catch are small. The "tail" of the net (the part that holds the big, heavy fish) is empty. The net is light and easy to handle.
Weak Light Tails: The net is mostly empty, but there's a tiny, tricky spot where a few heavy fish might get stuck. It's still mostly light, but you have to be careful.
No Light Tails: The net is heavy with big fish everywhere. It's too heavy to lift, and the math breaks down.

In the paper, a sieve has "Light Tails" if the "heavy" congruence classes (the ones that remove a lot of numbers) are so rare and far apart that they don't mess up the overall pattern.

The Main Discovery: The "Generic" Point

The paper proves a massive theorem (Theorem 1.1) that connects three things:

The Sieve is "Light-Tailed": The heavy fish aren't clogging the net.
The Pattern Exists: You can calculate exactly what percentage of numbers will fall through the sieve (the density).
The "Generic" Behavior: The set of numbers that fall through is a "generic point."

What does "Generic Point" mean?
Imagine a spinning roulette wheel with a million slots.

If the wheel is "generic," the ball will eventually visit every single slot in the exact proportion of how big the slots are. It doesn't get stuck in one corner.
Araújo proves that if your sieve has "Light Tails," the numbers that survive the sieve behave exactly like that ball on the roulette wheel. They are perfectly distributed and follow the laws of probability.

Why This Matters: The "Visible Lattice Points"

The paper isn't just about abstract numbers. It applies to real geometric problems.

The Analogy of the Forest:
Imagine you are standing in a forest of trees arranged in a perfect grid (a lattice). You look out into the distance.

The Problem: Some trees block your view of others. You can only see the "visible" trees (those not blocked by a tree closer to you).
The Math: This is the set of "visible lattice points."
The Result: Araújo shows that even in complex, multi-dimensional forests (algebraic number fields), the visible trees follow the same predictable, random patterns as the squarefree numbers.

The "Light Tail" vs. "Strong Light Tail" Distinction

The author makes a crucial distinction between two types of light tails:

Strong Light Tails: The heavy fish are completely gone. This happens naturally for the standard "B-free" systems (like the squarefree numbers). The math is clean and stable.
Weak Light Tails: The heavy fish are mostly gone, but there's a tiny chance they could appear if you look in a very specific, weird way. This is a more fragile state.

The Surprise:
The paper shows that for the standard "B-free" systems (like squarefree numbers), you always get Strong Light Tails. This means the standard systems are robust and stable. However, if you start tweaking the sieve (removing specific classes), you might drop into Weak Light Tails, where the math is still possible but much more delicate.

The Conclusion: A New Universal Rule

Francisco Araújo has successfully taken Sarnak's program—which was originally a set of rules for simple numbers—and expanded it to cover:

Complex Numbers: Working in "Étale Q-algebras" (which are like combinations of different number worlds).
Weird Sieves: Removing any pattern of numbers, not just multiples of squares.
Dynamic Behavior: Proving that these sets of numbers behave like a well-oiled machine (a dynamical system) that is predictable and random at the same time.

In a Nutshell:
The paper says: "As long as your sieve isn't too heavy (Light Tails), the numbers that survive will dance in a perfect, predictable rhythm, no matter how complex the dance floor is."

This unifies a century of number theory, showing that the deep, random beauty of prime numbers and squarefree numbers is a universal law that holds true even in the most exotic mathematical landscapes.

1. Problem Statement and Motivation

The paper aims to generalize Sarnak's Program, a set of conjectures and results concerning the dynamical properties of sets of "free" numbers (e.g., square-free integers), to a much broader class of number-theoretic sets known as Erdős sieves.

Context: Traditionally, Sarnak's program has been studied over the integers $\mathbb{Z}$ and specific number fields, focusing on $B$ -free systems where one removes multiples of a set of ideals $B$ (specifically, removing the congruence class $0 \pmod b$ for all $b \in B$ ).
Generalization: The author extends this framework to étale $\mathbb{Q}$ -algebras (finite products of number fields) and, crucially, to general sieves. A general sieve $R$ allows for the removal of arbitrary congruence classes modulo a set of pairwise coprime ideals, not just the zero class.
Core Question: Under what conditions do these generalized sets of "R-free numbers" ( $F_R$ $F_{R}$ ) exhibit the same topological and measure-theoretic dynamical behaviors as the classical square-free integers? Specifically, the paper seeks to establish:
1. The existence of a Mirsky measure for which $F_R$ is a generic point.
2. The calculation of topological entropy.
3. The structure of the admissible set space ( $\Omega_R$ ).
4. The existence of non-trivial joinings with group rotations.

2. Methodology and Framework

The paper employs a synthesis of Algebraic Number Theory and Topological/Measure-Theoretic Dynamics.

Mathematical Setting:
- Space: The ring of integers $\mathcal{O}_K$ of an étale $\mathbb{Q}$ -algebra $K$ .
- Sieve Definition: A sieve $R$ is a pair $(B_R, (R_b)_{b \in B_R})$ , where $B_R$ is a set of pairwise coprime invertible ideals, and $R_b$ is a finite set of congruence classes modulo $b$ .
- R-free Numbers: $F_R = \mathcal{O}_K \setminus \bigcup_{b \in B_R} R_b$ .
- Dynamical Systems: The study takes place in the shift space $\{0, 1\}^{\mathcal{O}_K}$ ${0, 1}^{O_{K}}$ equipped with the shift action $S_a(A) = A - a$ $S_{a} (A) = A - a$ . Two key spaces are analyzed:
  - $X_R$ : The orbit closure of $F_R$ .
  - $\Omega_R$ : The space of all $R$ -admissible sets (sets that do not cover the whole ring modulo any $b \in B_R$ ).
Key Concept: Light Tails:
The central methodological innovation is the definition of Light Tail conditions on the sieve $R$ . These conditions control the "tail" of the union of removed congruence classes as the index of the ideals grows.
- Weak Light Tails: The density of elements removed by new ideals (indices $> L$ ) that were not removed by previous ideals (indices $\le L$ ) vanishes as $L \to \infty$ .
- Strong Light Tails: The density of the union of all removed classes for indices $> L$ vanishes as $L \to \infty$ .
- Erdős Sieves: Sieves where $\sum_{b \in B_R} \frac{|R_b|}{N(b)} < \infty$ .
Tools:
- Mirsky Measure ( $\nu_R$ ): Defined as the pushforward of the Haar measure on the odometer group $G_R = \prod_{b \in B_R} \mathcal{O}_K/b$ via a factor map $\phi_R$ .
- Følner Sequences: Used to define densities and genericity in the non-compact setting of $\mathcal{O}_K$ .
- Chinese Remainder Theorem: Used extensively to analyze the structure of admissible sets and the odometer group.

3. Key Contributions and Results

A. Generalization of Sarnak's Point (1): Genericity and Density

The paper establishes a precise equivalence between the dynamical property of genericity and the number-theoretic property of light tails.

Theorem 3.21: For an Erdős sieve $R$ $R$ and a Følner sequence $I_N$ $I_{N}$ , the following are equivalent:
1. $F_R$ is a generic point for the system $(\Omega_R, S, \nu_R)$ with respect to $I_N$ .
2. $R$ has weak light tails with respect to $I_N$ .
3. The set $F_R$ has a well-defined density given by the product formula:
  $d_{I_N}(F_R) = \prod_{b \in B_R} \left(1 - \frac{|R_b|}{N(b)}\right)$
Significance: This proves that the existence of a natural density for the set of free numbers is not automatic for general sieves; it is strictly tied to the "weak light tails" condition. The paper provides counterexamples where the density does not exist or differs from the Mirsky measure prediction.

B. Generalization of Sarnak's Points (2), (4), and (5)

The paper extends other pillars of Sarnak's program to general sieves over étale algebras:

Entropy (Point 2): The topological entropy of the system $(\Omega_R, S)$ is calculated as:
$h_{top}(\Omega_R, S) = \log(2) \prod_{b \in B_R} \left(1 - \frac{|R_b|}{N(b)}\right)$
This generalizes the result for square-free integers ( $6/\pi^2$ ) to arbitrary sieves.
Proximality (Point 4): The system $(\Omega_R, S)$ is shown to be proximal. The empty set $\emptyset$ is the unique minimal subsystem (fixed point). Consequently, the system is not topologically weakly mixing.
Joinings (Point 5): The system $(\Omega_R, S)$ admits a non-trivial joining with the rotation system on the odometer group $(G_R, T)$ . This implies the system is not disjoint from the group rotation, a key feature in Sarnak's original program.

C. Analysis of Light Tail Properties

Section 5 provides a deep investigation into the stability and existence of light tail conditions:

Existence: Theorem 5.7 proves that all Erdős $B$ -free systems (where $R_b = b$ ) over any étale $\mathbb{Q}$ -algebra satisfy strong light tails for the standard Følner sequence $B_N$ . This generalizes previous results restricted to $\mathbb{Z}$ or specific number fields.
Weak vs. Strong: The paper demonstrates that Weak Light Tails $\nRightarrow$ Strong Light Tails.
- Example 5.14: A sieve is constructed that has weak light tails but fails strong light tails.
- Stability: Strong light tails are stable under reordering and adding/removing finite congruence classes. Weak light tails are not stable under removing congruence classes (Theorem 5.14).
Local-Global Principle: The paper distinguishes between the "local-global principle" (solvability of congruences) and light tails, showing they are not equivalent.

4. Significance and Impact

Unification of Number Theory and Dynamics: The paper successfully unifies the study of diverse number-theoretic sets (square-free numbers, visible lattice points, values of polynomials, etc.) under a single dynamical framework (Erdős sieves over étale algebras).
Resolution of Density Questions: It clarifies exactly when the density of $R$ -free numbers exists and equals the probabilistic prediction (the product formula). It shows that for general sieves, this is a non-trivial condition (weak light tails) rather than an inherent property.
Foundations for Part II: This paper lays the necessary groundwork (defining the spaces, measures, and tail conditions) for Part II of the series, which will address the isomorphism of the system to a minimal rotation (Sarnak's Point 3) and specific applications (e.g., square-free values of polynomials).
Counterexamples and Nuance: By constructing sieves where density fails or where weak/strong light tails diverge, the paper highlights the fragility of these properties in general settings, preventing over-generalization of results known for $\mathbb{Z}$ .

In summary, this paper provides a rigorous topological and measure-theoretic foundation for the study of generalized sieves, proving that the "Sarnak Program" holds for a vast class of number-theoretic sets provided they satisfy specific "light tail" conditions, and characterizing the dynamical structure of these sets in detail.