Wiener's lemma along primes and other subsequences

TL;DR

This paper investigates Wiener's lemma and the extremal behavior of measures on the unit circle along various subsequences like primes and polynomials, with implications for ergodic theory and operator dynamics.

Contribution

It extends Wiener's lemma to arithmetic subsequences such as primes and polynomials, connecting it to ergodic and operator theory, and discusses open problems and research directions.

Findings

Validity of Wiener's lemma along primes and polynomials analyzed

Connections established between subsequence behavior and operator orbits

Open questions posed regarding polynomial averages and Wiener-Wintner results

Abstract

Inspired by subsequential ergodic theorems, we study the validity of Wiener's lemma and the extremal behavior of a measure $\mu$ on the unit circle via the behavior of its Fourier coefficients $\hat\mu(k_n)$ along subsequences $(k_n)$. We focus on arithmetic subsequences such as polynomials, primes and polynomials of primes, and also discuss connections to rigidity sequences, return times sequences and strongly sweeping out sequences as well as measures on $\mathbb{R}$. We also present consequences for orbits of operators and of $C_0$-semigroups on Hilbert and Banach spaces extending the results of Goldstein and Goldstein, Nagy. The results are complemented by some open questions and indication of interesting research directions. After this paper had been published, it was pointed out to us by Emmanuel Lesigne and M\'at\'e Wierdl that there is a gap in the Example on return times sequences along polynomials on page 13. Indeed, to make the argument there work, one needs a Wiener-Wintner type result for polynomial averages with a precise information about the limit, and this is presently out of reach.