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The Wiener Wintner and Return Times Theorem Along the Primes

This paper establishes the first extension of the Wiener-Wintner Theorem to arithmetic sequences by proving the almost everywhere convergence of weighted ergodic averages along prime times for $L^p$ functions, utilizing a novel synthesis of classical Fourier analysis, combinatorial number theory, and higher-order Fourier analysis with significant $U^3$ -estimates for Heath-Brown models of the von Mangoldt function.

Original authors: Jan Fornal, Anastasios Fragkos, Ben Krause, Michael Lacey, Hamed Mousavi, Yu-Chen Sun

Published 2026-07-07

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CC BY 4.0

Original authors: Jan Fornal, Anastasios Fragkos, Ben Krause, Michael Lacey, Hamed Mousavi, Yu-Chen Sun

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 watching a complex, chaotic dance. In this dance, a group of people (representing a mathematical system) move around a room according to strict rules. Every time a person moves, they leave a mark. If you watch long enough, you can predict the general pattern of where they will be. This is the basic idea of Ergodic Theory, a branch of mathematics that studies how systems evolve over time.

For decades, mathematicians have known a rule called the Wiener-Wintner Theorem. Think of this theorem as a guarantee that no matter how you "tune" your observation (like changing the pitch of a sound or the color of a light), the dance will eventually settle into a predictable rhythm.

However, there was a missing piece. The original theorem worked for every step in the dance (1, 2, 3, 4...). But what if you only watched the dancers at specific, irregular intervals? Specifically, what if you only looked at the steps numbered 2, 3, 5, 7, 11, 13...? These are the prime numbers. They are the "wild cards" of mathematics—irregular, hard to predict, and scattered in a way that doesn't follow a simple pattern.

This paper, by Fornal, Fragkos, Krause, Lacey, Mousavi, and Sun, proves that the Wiener-Wintner Theorem still works even if you only watch the prime-numbered steps.

Here is a breakdown of how they did it, using simple analogies:

1. The Problem: The "Prime" Puzzle

The authors wanted to prove that if you watch a system only at prime times ( $p_1, p_2, p_3...$ ), the average behavior still converges to a stable result, regardless of how you "tune" your observation (represented by the variable $\theta$ ).

The difficulty is that prime numbers are notoriously difficult to handle. They don't follow a smooth curve; they are jagged and unpredictable. Trying to analyze them directly is like trying to predict the weather by only looking at the sky on days that are prime numbers.

2. The Solution: The "Heath-Brown Model" (The Proxy)

Since primes are so messy, the authors didn't try to study the primes directly. Instead, they used a proxy.

Imagine you want to study the behavior of a wild, untamed horse (the prime numbers). It's too dangerous to get close. So, you build a very accurate, high-tech robot horse (the Heath-Brown Model) that mimics the real horse's movements perfectly in almost every way.

In mathematics, this "robot horse" is an approximation of the von Mangoldt function (a tool that highlights prime numbers). The authors used a specific version of this model created by mathematician Heath-Brown.

3. The Secret Weapon: The "U3 Norm" (The Stress Test)

To prove their robot horse is a good enough substitute, they had to prove it behaves exactly like the real horse in a very specific, high-stress environment. They used a tool called the Gowers $U^3$ Norm.

Think of the $U^3$ Norm as a stress test or a complexity meter.

If you take a simple, boring pattern (like a straight line), the stress test gives it a low score.
If you take a chaotic, random mess, it gives it a high score.
The authors needed to show that their "robot horse" (the Heath-Brown model) has a very low stress score when compared to the real primes.

They proved two crucial things:

The Robot is Close: The Heath-Brown model is so close to the real prime numbers that, for the purpose of their dance, they are indistinguishable.
The Robot is Simple: The specific parts of the robot model that are "fixed" (not changing) are surprisingly simple and predictable. They proved that the "complexity" of these fixed parts drops off very quickly (a bound of $Q^{-3/8}$ ), which is much better than anyone expected.

4. The "Nilsequence" (The Hidden Structure)

The authors also had to deal with the "structure" of the dance itself. They used advanced tools from Higher Order Fourier Analysis.

Imagine the dance floor has invisible, complex gears underneath it. Sometimes, the dancers move in a way that looks random, but it's actually driven by these hidden gears. The authors used a theorem (the Inverse Theorem) to show that if the dance doesn't settle down, it must be because of these hidden gears (called nilsequences).

They then showed that the "prime number dance" doesn't get stuck on these hidden gears. Because the "robot horse" (the model) is so simple and well-behaved, it can't get trapped in the complex gears that would ruin the prediction.

5. The Final Result

By combining these ideas, the authors proved:

You can watch the system only at prime-numbered times.
You can tune your observation in any way you like (any frequency $\theta$ ).
The average of what you see will always settle down to a specific value for almost every starting point in the system.

Summary

The paper is a triumph of substitution and simplification.

The Problem: Primes are too chaotic to study directly for this specific theorem.
The Trick: Replace the chaotic primes with a "robot" (Heath-Brown model) that acts like them.
The Proof: Show that the robot is simple enough (low $U^3$ norm) that it can't fool the system into being unpredictable.
The Conclusion: The rules of the dance hold true, even if you only watch the prime-numbered steps.

This is the first time this specific "Wiener-Wintner" rule has been extended to an arithmetic sequence (like the primes), bridging the gap between the smooth world of classical analysis and the jagged, irregular world of number theory.

Technical Summary: The Wiener-Wintner Theorem Along the Primes

Problem Statement
The paper addresses the extension of the classical Wiener-Wintner Ergodic Theorem to arithmetic sequences, specifically the sequence of prime numbers. The classical theorem (Theorem 1.1) asserts that for a measure-preserving system $(X, \nu, T)$ and a function $f \in L^1(X)$ , there exists a set of full measure such that the weighted averages $\frac{1}{N} \sum_{n \le N} \phi(n) f(T^n x)$ converge for all continuous 1-periodic functions $\phi$ . While it is known that the limit exists for almost every $\theta$ when $\phi(n) = e^{2\pi i n \theta}$ (via Birkhoff's theorem applied to the product system), the Wiener-Wintner theorem strengthens this by asserting convergence for all $\theta$ simultaneously on a single full measure set.

The authors aim to prove Theorem 1.2: for any measure-preserving system and $f \in L^\infty(X)$ (relaxable to $L^p$ for $p>1$ ), there exists a full measure set $X_f$ such that for all $x \in X_f$ , the averages
$\lim_{N \to \infty} \frac{1}{N} \sum_{n \le N} \phi(p_n) f(T^{p_n} x)$
converge for all continuous 1-periodic $\phi$ , where $\{p_n\}$ are the prime integers. This represents the first extension of the Wiener-Wintner theorem to an arithmetic sequence.

Methodology
The proof operates at the intersection of classical Fourier analysis, combinatorial number theory, higher-order Fourier analysis, and pointwise ergodic theory. The methodology relies on three primary pillars:

Approximation of the von Mangoldt Function:
The authors utilize the Heath-Brown model ( $\Lambda_{HB}$ ) to approximate the von Mangoldt function $\Lambda(n)$ . Specifically, they employ a truncated sum $\Lambda_{\le T}$ defined by dyadic sums of terms $\Lambda_Q(n)$ , where $\Lambda_Q$ involves Ramanujan sums $c_q(n)$ and the Möbius function $\mu(q)$ .
- Siegel Zero Correction: To handle potential exceptional zeros of Dirichlet $L$ -functions, the model is refined to $\Lambda'_{\le T}$ by incorporating a correction term involving the exceptional character $\chi_{q_{Siegel}}$ and the Siegel zero $\sigma$ .
Higher-Order Fourier Analysis (Gowers Norms):
The core technical engine involves establishing bounds on the Gowers $U^3$ norms.
- Approximation Error: The authors prove that the Heath-Brown model is close to the von Mangoldt function in the $U^3$ norm: $\|\Lambda - \Lambda'_{\le Q}\|_{U^3[N]} \lesssim (\log N)^{-A}$ . This is achieved using the quantitative inverse theorem for the $U^3$ norm (due to Leng, Sanders, Green, and Tao), which correlates functions with large $U^3$ norms to nilsequences. The proof involves decomposing $\Lambda$ and $\Lambda'$ into Type I, Type II, and twisted Type I sums and showing that none can correlate significantly with the structured nilsequences unless a contradiction arises.
- Fixed Complexity Bounds: A crucial combinatorial result (Proposition 2.1) establishes a sharp bound for the $U^3$ norm of the fixed-complexity component $\Lambda_Q$ :
  $\|\Lambda_Q\|_{U^3[N]} \lesssim Q^{-3/8 + o(1)}$
  This bound is derived through elementary but non-trivial combinatorial arguments involving the structure of linear forms over the cube $\{0,1\}^3$ and divisibility constraints on the denominators of the Ramanujan sums. The authors note that a weaker $Q^{-1/4}$ bound is accessible, but the $3/8$ exponent is essential for their proof.
Ergodic Transference and Uniformity:
The proof utilizes a transference argument to move from the weighted averages with $\Lambda$ to the prime averages.
- Weak Mixing Reduction: By the ergodic decomposition, the problem reduces to the case where $f$ is weakly mixing. For weakly mixing functions, the uniform Wiener-Wintner theorem (equation 1.6) implies that the supremum over $\theta$ of the unweighted averages vanishes.
- Weighted Averages: The authors combine the $U^3$ estimates with inequalities relating Wiener-Wintner averages to the $U^3$ norm of the weight (Proposition 1.11). Specifically, they show that if the weight has a small $U^3$ norm, the weighted averages are small. By approximating $\Lambda$ with $\Lambda_{\le Q}$ (which has small $U^3$ norm for large $Q$ ) and controlling the error, they establish the convergence of the prime averages.

Key Contributions and Results

Theorem 1.2: The primary result is the proof of the Wiener-Wintner theorem along the primes. It guarantees the existence of a full measure set where the weighted prime averages converge for all continuous periodic weights.
Proposition 2.1 (Fixed Complexity $U^3$ Bound): The paper establishes the bound $\|\Lambda_Q\|_{U^3[N]} \lesssim Q^{-3/8 + o(1)}$ for $N > Q^{20}$ . This is a significant improvement over the standard $Q^{-1/4}$ bound and is noted as a result of independent interest. The proof relies on a detailed combinatorial analysis of the constraints imposed by the linear forms in the Gowers norm expansion.
Proposition 3.1 (Heath-Brown Approximation): The paper proves that the Heath-Brown model (with Siegel zero correction) approximates the von Mangoldt function in the $U^3$ norm with error decaying as a power of $\log N$ . This relies on the inverse $U^3$ theorem and a detailed analysis of Type I/II sums and nilsequences.
Lemma 3.19 and 3.23: These lemmas provide estimates for the $U^3$ norms of the Heath-Brown model twisted by exceptional characters, handling the technical difficulties arising from potential Siegel zeros.

Significance
The paper claims to provide the first extension of the Wiener-Wintner theorem to arithmetic sequences. The significance lies in:

Unifying Fields: The proof synthesizes tools from higher-order Fourier analysis (specifically $U^3$ theory and nilsequences) with classical analytic number theory (Heath-Brown models, Siegel zeros) and ergodic theory.
Combinatorial Insight: The derivation of the $Q^{-3/8}$ bound for the fixed-complexity Heath-Brown model is highlighted as a non-trivial combinatorial achievement that surpasses standard dimension-counting bounds.
Robustness of Prime Averages: The result demonstrates that the prime numbers, despite their irregular distribution, possess sufficient "randomness" (in the sense of small Gowers norms) to satisfy strong pointwise ergodic theorems when weighted by continuous functions.

The authors explicitly state that their $U^3$ estimates for the Heath-Brown models may be of independent interest to researchers in additive combinatorics and analytic number theory. The work does not propose new experimental applications but rather solidifies the theoretical foundation for pointwise convergence along primes in dynamical systems.