Wiener's lemma along primes and other subsequences

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 Máté 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.

34 pages; some typos corrected; a gap in Theorem 5.5 (i) <=> (iii) (previous version) is resolved by removing that part and formulating a consequence for prime powers of operators (Corollary 5.6 in the current version). More examples of extremal sequences arising from ergodic theory have been added. Some remarks about connections to strong sweeping out sequences have been added