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The Wiener Wintner and Return Times Theorem Along the Primes

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

TL;DR

The paper resolves extending the Wiener-Wintner and Return Times theorems to averages taken at prime times, establishing almost-sure convergence for weights and return-time correlations along $p_n$. The authors fuse classical Fourier analysis, higher-order Fourier techniques, and combinatorial number theory, centering on $U^3$ control via Heath-Brown models of the von Mangoldt function to handle sparsity along the primes. Key technical contributions include sharp $U^3$ bounds for fixed-complexity HB components and $U^3$-type approximation results for the von Mangoldt function, underpinning uniform Wiener-Wintner and Return Times arguments along prime indices. The results deepen the connection between ergodic theory and analytic number theory for sparse time sequences and provide a framework for further extensions to other sparse sets or polynomial modulations.

Abstract

We prove the following Return Times Theorem along the sequence of prime times, the first extension of the Return Times Theorem to arithmetic sequences: For every probability space, $(Ω,ν)$, equipped with a measure-preserving transformation, $T \colon Ω\to Ω$, and every $f \in L^\infty(Ω)$, there exists a set of full probability, $Ω_f \subset Ω$ with $ν(Ω_f) =1$, so that for all $ω\in Ω_f$, for any other probability space $(X,μ)$, equipped with a measure-preserving transformation $S : X \to X$, for any $g \in L^{\infty}(X)$, \begin{align} \frac{1}{N} \sum_{n \leq N} f(T^{p_n} ω) g(S^{p_n} \cdot) \end{align} converges $μ$-almost surely; above, $\{ 2=p_1 < p_2 < \dots \}$ are an enumeration of the primes. The Wiener-Wintner theorem along the primes is an immediate corollary. Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with $U^3$ theory playing an important role; our $U^3$-estimates for \emph{Heath-Brown} models of the von Mangoldt function may be of independent interest.

The Wiener Wintner and Return Times Theorem Along the Primes

TL;DR

The paper resolves extending the Wiener-Wintner and Return Times theorems to averages taken at prime times, establishing almost-sure convergence for weights and return-time correlations along . The authors fuse classical Fourier analysis, higher-order Fourier techniques, and combinatorial number theory, centering on control via Heath-Brown models of the von Mangoldt function to handle sparsity along the primes. Key technical contributions include sharp bounds for fixed-complexity HB components and -type approximation results for the von Mangoldt function, underpinning uniform Wiener-Wintner and Return Times arguments along prime indices. The results deepen the connection between ergodic theory and analytic number theory for sparse time sequences and provide a framework for further extensions to other sparse sets or polynomial modulations.

Abstract

We prove the following Return Times Theorem along the sequence of prime times, the first extension of the Return Times Theorem to arithmetic sequences: For every probability space, , equipped with a measure-preserving transformation, , and every , there exists a set of full probability, with , so that for all , for any other probability space , equipped with a measure-preserving transformation , for any , \begin{align} \frac{1}{N} \sum_{n \leq N} f(T^{p_n} ω) g(S^{p_n} \cdot) \end{align} converges -almost surely; above, are an enumeration of the primes. The Wiener-Wintner theorem along the primes is an immediate corollary. Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with theory playing an important role; our -estimates for \emph{Heath-Brown} models of the von Mangoldt function may be of independent interest.
Paper Structure (9 sections, 17 theorems, 211 equations, 2 figures)

This paper contains 9 sections, 17 theorems, 211 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notation
  4. Gowers Norms
  5. $U^3$ Norms of the Heath-Brown Model, with Fixed Complexity
  6. $U^3$ Norms of the Von Mangoldt Function and the Heath-Brown Model
  7. Proof of the Wiener-Wintner Theorem
  8. Proof of the Return Times Theorem
  9. Approximation of the von Mangoldt Function in Arithmetic Progressions

Key Result

Theorem 1.1

Let $(X,\nu,T)$ be a measure-preserving system, and let $f \in L^1(X)$ be arbitrary. Then there exists a subset $X_f \subset X$ with $\mu(X_f) = 1$ so that for all $x \in X_f$ exists for all continuous $1$-periodic $\phi \colon \mathbb R\to\mathbb C$.

Figures (2)

  • Figure 1: An example of labelling of a cube with green and blue marked vertices in the case $|S|=6$.
  • Figure 2: The four cases, up to symmetry, in which there are 5 marked vertices on the cube, with 1 being green, and no face having 4 marked vertices.

Theorems & Definitions (32)

  • Theorem 1.1: Wiener-Wintner Ergodic Theorem
  • Theorem 1.3: Return Times Theorem
  • Theorem 1.4
  • Lemma 1.26: Gowers-Cauchy-Schwarz Inequality
  • Proposition 1.31
  • proof
  • Remark 1.53
  • proof
  • Proposition 2.1
  • Remark 2.3
  • ...and 22 more