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

This paper generalizes Sarnak's program to Erdős sieves by establishing a connection between the dynamical systems of $R$ -free algebraic integers and $R$ -admissible sets, proving the latter is isomorphic to an ergodic rotation, and applying these results to infinite sumsets, squarefree polynomial values, and a prime number theorem for $R$ -free numbers.

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

Imagine you are a master gardener trying to grow a very specific type of rare flower. You have a giant field (representing all the numbers in the universe), but you need to remove certain patches of soil where weeds grow.

In mathematics, this process is called sieving. You take a sieve (a set of rules) and shake it over your field to remove the "bad" numbers (the weeds) and keep the "good" ones (the flowers).

This paper, written by Francisco Araújo, is the second part of a deep investigation into a specific, very complex type of gardening called Erdős Sieves. It builds on a famous idea by mathematician Peter Sarnak. Here is the story of the paper, broken down into simple concepts.

1. The Garden and the Rules (The Sieve)

Imagine your field is the set of all integers (1, 2, 3...).

The Weeds: Usually, we remove numbers divisible by squares (like 4, 9, 16) to find "square-free" numbers. This is the classic sieve.
The New Rules: This paper looks at much more complicated rules. Instead of just removing multiples of 4, maybe you remove numbers that leave a remainder of 1 when divided by 4, OR numbers that leave a remainder of 3 when divided by 9, and so on, using an infinite list of different rules.
The Goal: We want to know what the "good" numbers (the ones that survive the sieve) look like. Do they form a nice, predictable pattern? Or are they chaotic?

2. Two Ways to Look at the Garden

The author introduces two different ways to study these surviving numbers, which he calls $X_R$ and $\Omega_R$ .

$X_R$ (The Actual Garden): This is the set of all the actual surviving numbers you find in your field. It's the "real world" data.
$\Omega_R$ (The Blueprint): This is a theoretical "blueprint" or a library of all possible patterns that could survive your rules, even if they don't actually appear in your specific field right now.

The Big Question: Do the actual garden ( $X_R$ ) and the theoretical blueprint ( $\Omega_R$ ) look the same?

In simple sieves (like just removing multiples of 4), they are identical.
In these complex "Erdős Sieves," they are often different. The blueprint might contain patterns that the actual garden never produces.

3. The "Light Tail" Analogy

To understand when the garden and the blueprint match, the author uses a concept called "Light Tails."

Imagine your sieve rules are like a long line of people passing a bucket of water.

Heavy Tails: If the people at the very end of the line (the infinite part of the rules) are still carrying huge buckets of water, they splash everywhere. This messes up the garden, making the actual garden ( $X_R$ ) look very different from the blueprint ( $\Omega_R$ ).
Light Tails: If the people at the end of the line are carrying tiny, almost empty cups, their splash is negligible. The water settles nicely.
The Discovery: The author proves that if your rules have "Light Tails" (the rules get weaker and weaker as you go further out), then the actual garden and the blueprint do match up in a statistical sense. The garden fills up the blueprint perfectly.

4. The "Minimal" Sieve (The Ultimate Filter)

Sometimes, your sieve rules are redundant.

Example: You might have a rule saying "Remove multiples of 4" and another saying "Remove multiples of 2." But removing multiples of 2 automatically removes multiples of 4!
The author introduces the idea of a "Minimal Sieve." This is like stripping your filter down to its absolute bare bones. You remove every redundant rule until you have the smallest, most efficient set of rules that produces the exact same result.
The Result: He proves that for any complex sieve, there is a unique "minimal" version (under certain conditions) that acts as the "true identity" of that sieve.

5. The Magic Machine (Dynamical Systems)

This is the most abstract part, but here is the metaphor:
The author shows that the behavior of these surviving numbers is exactly the same as a rotating wheel on a giant, smooth, circular track.

Imagine a wheel with a pattern painted on it. As the wheel spins, the pattern moves.
The author proves that the "surviving numbers" move through the universe in a way that is mathematically identical to this spinning wheel.
Why does this matter? Because we know a lot about spinning wheels (they are predictable, they have specific frequencies). By proving the numbers act like a wheel, the author can predict how the numbers behave, how they are distributed, and how they interact with other mathematical objects.

6. Real-World Applications (The Harvest)

Finally, the author uses these tools to solve real number theory problems:

Infinite Sums: He shows that if you take a set of "good" numbers (like square-free numbers), you can always find two infinite groups of numbers that, when added together, stay inside your "good" set. It's like saying you can always find two groups of friends who, when they mix, only hang out with other friends who follow the same rules.
Polynomials: He applies this to polynomials (like $x^2 + 1$ ). He proves that the values of these polynomials that are "square-free" behave very predictably, just like the spinning wheel.
Prime Number Theorem: He derives a new version of the famous Prime Number Theorem. This theorem helps us count how many prime numbers (or "prime-like" numbers) exist in a certain range. His method works for these complex sieves, not just for standard primes.

Summary

In simple terms, this paper is about finding order in chaos.
Mathematicians have a set of complex rules for filtering numbers. This paper proves that if those rules get "lighter" as you go further out, the resulting numbers behave in a very orderly, predictable way (like a spinning wheel). It also gives us a way to simplify any complex rule down to its simplest form and use that to solve problems about how numbers are distributed in the universe.

1. Problem Statement and Context

This paper is the second part of a series generalizing Sarnak's Program (originally formulated for squarefree numbers) to a broader class of number-theoretic sets known as Erdős sieves.

The Setting: The authors work within the ring of integers $\mathcal{O}_K$ of an étale $\mathbb{Q}$ -algebra $K$ (a finite product of number fields).
The Object of Study: An Erdős sieve $R$ is defined by a collection of congruence classes $R_b$ modulo an infinite set of pairwise coprime ideals $B_R$ . The set of $R$ -free numbers, denoted $F_R$ , consists of elements in $\mathcal{O}_K$ that do not fall into any of the forbidden congruence classes $R_b$ .
The Core Question: The paper investigates the dynamical systems associated with these sets. Specifically, it compares two distinct spaces:
1. $X_R$ : The orbit closure of the set $F_R$ in the shift space $\{0,1\}^{\mathcal{O}_K}$ . This represents the topological dynamics of the set.
2. $\Omega_R$ : The set of all $R$ -admissible sets (sets $A$ such that $-A + R_b \neq \mathcal{O}_K$ for all $b$ ). This represents the measure-theoretic constraints.
The Gap: For the classical squarefree sieve, it is known that $X_R = \Omega_R$ . However, for general Erdős sieves, this equality often fails. The paper aims to characterize the relationship between $X_R$ and $\Omega_R$ , compute the spectral properties of the associated dynamical systems, and derive number-theoretic applications.

2. Methodology

The authors employ a synthesis of ergodic theory, dynamical systems, and analytic number theory.

Minimal Sieves: A central methodological innovation is the introduction of minimal sieves. The authors define operations of dilation (combining congruence classes) and contraction (splitting them). Every sieve is shown to be equivalent to a "minimal" sieve where the ideals in the support cannot be further decomposed. This allows for a unique characterization of the dynamical system up to isomorphism.
Measure-Theoretic Isomorphism: The authors construct a compact abelian group $G_{R,F}$ associated with the sieve. They prove that the dynamical system $(\Omega_R, S, \nu_R)$ (where $\nu_R$ is the Mirsky measure) is isomorphic to a rotation on this group.
Light Tail Conditions: The analysis relies heavily on distinguishing between weak light tails and strong light tails.
- Weak light tails: The density of the "tail" of the sieve (elements removed by large ideals but not small ones) vanishes.
- Strong light tails: The density of the union of all large ideals vanishes.
  These conditions determine whether the set $F_R$ is a generic point for the measure $\nu_R$ .
Spectral Analysis: Using the Halmos-von Neumann theorem, the authors compute the point spectrum of the system by analyzing the characters of the group $G_{R,F}$ .

3. Key Contributions and Results

A. Relationship between $X_R$ and $\Omega_R$

Theorem 1.1: For an Erdős sieve $R$ , there exists a Følner sequence with respect to which $R$ has weak light tails if and only if the Mirsky measure of the orbit closure is full, i.e., $\nu_R(X_R) = 1$ .
Theorem 5.8: If $R$ has weak light tails, then the hereditary hull of $X_R$ (denoted $\tilde{X}_R$ ) is equal to $\Omega_R$ .
Theorem 5.11: For minimal Erdős sieves with weak light tails, the following are equivalent:
1. $X_R = X_{R'}$
2. $\Omega_R = \Omega_{R'}$
3. The sieves are equivalent up to a shift (same support $B_R = B_{R'}$ and $R_b = \delta_b + R'_b$ ).
  Significance: This establishes that for "well-behaved" sieves, the topological structure of the orbit closure is entirely determined by the measure-theoretic admissible sets.

B. Isomorphism to Group Rotations

Theorem 1.4 (Main Structural Result): The system $(\Omega_R, S, \nu_R)$ is isomorphic to a rotation $(G_{R,F}, T^F, P_F)$ on a compact abelian group $G_{R,F} = \prod_{b \in B_R} \mathcal{O}_K / F(R_b)$ , where $F(R_b)$ is the stabilizer subgroup of the congruence class $R_b$ .
Spectrum Computation (Theorem 4.19): The point spectrum of the system is explicitly computed as the set of characters $\chi$ of $\mathcal{O}_K$ that are trivial on the intersection of finitely many stabilizers $F(R_b)$ .
Uniqueness: The authors show that for minimal sieves over $\mathbb{Q}$ , the dynamical system is determined solely by the set of prime powers in the support, not the specific congruence classes chosen.

C. Number Theoretic Applications

Infinite Sumsets (Theorem 1.5): The paper proves that for sieves with strong light tails, the set of $R$ -free numbers $F_R$ contains infinite sumsets $A+B$ where $B$ has positive density. This addresses questions regarding the structure of sets with positive density that contain infinite sumsets.
Squarefree Values of Polynomials: The framework is applied to the set of squarefree values of polynomials $f(x)$ . The authors show that if $f$ is irreducible and the associated sieve has weak light tails, then $X_{R_f} = \Omega_{R_f}$ . They also provide density formulas for simultaneous squarefree values of multiple polynomials.
Ergodic Prime Number Theorem (Theorem 1.6 & 6.20): The paper generalizes the Prime Number Theorem to $R$ -free numbers. It proves that for any uniquely ergodic dynamical system $(X, T)$ and continuous function $f$ :
$\lim_{N \to \infty} \frac{1}{N} \sum_{m \in F_R \cap [1, N]} f(T^{\Omega(m)} x) = d(F_R) \int_X f d\mu$
where $\Omega(m)$ is the number of prime factors of $m$ . This result unifies and generalizes previous results by Moreira, Richter, Robertson, and others.

4. Significance

Unification: The paper successfully unifies the study of various "free" sets (squarefree, cubefree, carefree couples, polynomial values) under a single dynamical framework (Erdős sieves).
Resolution of Sarnak's Program: It completes the generalization of Sarnak's program by clarifying the precise conditions (weak light tails) under which the topological and measure-theoretic descriptions of these sets coincide.
New Tools: The introduction of minimal sieves and the contraction/dilation operations provides a robust algebraic toolset for comparing different sieves and determining their dynamical equivalence.
Applications: The results provide rigorous proofs for the existence of infinite sumsets within $R$ -free numbers and offer a powerful method for computing densities and asymptotic behavior of complex number-theoretic sets using ergodic theory.

In summary, this paper bridges the gap between the combinatorial structure of sieves and the spectral theory of dynamical systems, providing a comprehensive classification of Erdős sieves and their associated number-theoretic properties.