The Wiener Wintner Theorem Along the Primes

Abstract

We prove the following Wiener-Wintner Theorem along the sequence of prime times, the first extension of the Wiener-Wintner Theorem to arithmetic sequences: for every probability space, $(X,\hat{I}½),$ equipped with a measure-preserving transformation, $T:X\to X,$ and every $f\in L^p(X),1<p\le \infty$, there exists a set of full probability, $X_f\subset X$ with $\hat{I}½(X_f)=1,$ so that for all $\ddot{I}‰\in X_f$, \[ \frac{1}{N} \sum_{n \leq N} e^{ 2 πi p_n θ} f(T^{p_n} ω) \] converges for all $\hat{I}¸\in [0,1]$; above, $\{2=p_1<p_2<\dots \}$ are an enumeration of the primes. 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 Heath-Brown models of the von Mangoldt function may be of independent interest.