Wiener-Pitt sets for compact Abelian groups

TL;DR

This paper demonstrates that certain strongly continuous measures on compact Abelian groups, with Fourier transforms confined to rapidly decreasing sequences, have convolution squares that are absolutely continuous, thus avoiding the Wiener-Pitt phenomenon.

Contribution

It establishes conditions under which measures on compact Abelian groups do not exhibit the Wiener-Pitt phenomenon, extending previous research.

Findings

Measures with Fourier transforms in rapidly decreasing sequences have convolution squares absolutely continuous.

Strongly continuous measures with bounded norm and specific Fourier support avoid Wiener-Pitt phenomenon.

Results extend understanding of measure convolution behavior on compact Abelian groups.

Abstract

Suppose that $G$ is a compact Hausdorff Abelian group. We say $\mu \in M(G)$ is strongly continuous if $|\mu|(x+H)=0$ 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 $\mu\in M(G)$ with $\|\mu\| \leq 1$ and $\widehat{\mu}(\widehat{G})\subset \{a_n: n \in \mathbb{N}\}\cup\{0\}$, the measure $\mu\ast\mu$ is absolutely continuous with respect to Haar measure on $G$. This implies that $\mu$ does not exhibit the so-called Wiener-Pitt phenomenon. The paper is a continuation of investigations started in \cite{ow}.