Wiener-Pitt sets for compact Abelian groups
Przemysław Ohrysko, Tom Sanders, Michał Wojciechowski
TL;DR
This work investigates when a strongly continuous measure $\mu$ on a compact Abelian group $G$ has natural spectrum, focusing on the case where $\widehat{\mu}(\widehat{G})\subset\{a_n:n\in\mathbb{N}\}\cup\{0\}$ with $(a_n)$ a rapidly decaying sequence. In the torsion-free dual setting, the authors prove that if $a_1\le1$ and $a_{n+1}\le a_n\exp(-\exp(C a_n^{-6}))$ for a fixed constant $C>1$, then $\mu*\mu\in L^1(G)$ for every strongly continuous $\mu$ with $\|\mu\|\le 1$, hence $\mu$ has natural spectrum (and Wiener-Pitt phenomena do not occur for this class). The argument combines the Bo\v{z}ejko-P\l{}czynski uniform invariant approximation property (BPB), a spectral-layer decomposition, and a limiting procedure to obtain the $L^1$-limit corresponding to $\widehat{\mu}(\gamma)^2$. For the general case, the paper develops a finite-group quantitative framework via a refined Cohen idempotent theorem and then extends the result to arbitrary compact groups by a limiting argument, yielding the same natural-spectrum conclusion under the prescribed spectral constraint. Overall, the results provide explicit decay conditions on the Fourier-Stieltjes range that preclude Wiener-Pitt phenomena and extend prior work to a broad class of compact abelian groups.
Abstract
Suppose that $G$ is a compact Hausdorff Abelian group. We say $μ\in M(G)$ is strongly continuous if $|μ|(x+H)=0$ for any $x \in G$ and any $H \leq G$ that is closed and of infinite index. We prove that for any sufficiently rapidly decreasing sequence $(a_{n})_{n=1}^{\infty}\in c_{0}(\mathbb{N})$, for every strongly continuous $μ\in M(G)$ with $\|μ\| \leq 1$ and $\widehatμ(\widehat{G})\subset \{a_n: n \in \mathbb{N}\}\cup\{0\}$, the measure $μ\astμ$ is absolutely continuous with respect to Haar measure on $G$. This implies that $μ$ does not exhibit the so-called Wiener-Pitt phenomenon. The paper is a continuation of investigations started in \cite{ow}.