The Wiener Wintner and Return Times Theorem Along the Primes

Summary · qwen2.5:32b

The article extends the Wiener-Wintner Theorem to arithmetic sequences specifically along prime times, proving convergence for a given measure-preserving transformation and functions in $L^p(X)$ spaces. This extension bridges classical Fourier analysis with combinatorial number theory and ergodic theory, utilizing U^3 theory in its proof, which includes novel $U^3$-estimates for Heath-Brown models of the von Mangoldt function.

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Discovering the Wiener-Wintner Theorem's extension to prime times and its intersection with Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory. Check out this new proof for a fascinating mathematical journey!

Excerpt

arXiv:2601.10459v2 Announce Type: replace 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, $(X, \nu),$ equipped with a measure-preserving transformation, $T : X \to X,$ and every $f \in L^p(X), 1 < p \leq \infty$, there exists a set of full probability, $X_f \subset X$ with $\nu(X_f) = 1,$ so that for all $\omega \in X_f$, \[ \frac{1}{N} \sum_{n \leq N} e^{ 2 \pi i n \theta} f(T^{p_n} \omega) \] converges for all $\theta \in [0,1]$; 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.