The Wiener Wintner and Return Times Theorem Along the Primes

TL;DR

This paper extends the Return Times Theorem to prime times, showing convergence of certain averages along primes in measure-preserving systems, and connects Fourier analysis, number theory, and ergodic theory.

Contribution

It is the first to prove the Return Times Theorem along primes, combining techniques from Fourier analysis, number theory, and ergodic theory.

Findings

Proves convergence of averages along primes in measure-preserving systems.

Establishes a Wiener-Wintner type theorem along primes as a corollary.

Develops $U^3$-estimates for Heath-Brown models of the von Mangoldt function.

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, $(\Omega,\nu)$, equipped with a measure-preserving transformation, $T \colon \Omega \to \Omega$, and every $f \in L^\infty(\Omega)$, there exists a set of full probability, $\Omega_f \subset \Omega$ with $\nu(\Omega_f) =1$, so that for all $\omega \in \Omega_f$, for any other probability space $(X,\mu)$, 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} \omega) g(S^{p_n} \cdot) \end{align} converges $\mu$-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.