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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2510.24578 |
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| _version_ | 1866914118851624960 |
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| author | Ohrysko, Przemysław Sanders, Tom Wojciechowski, Michał |
| author_facet | Ohrysko, Przemysław Sanders, Tom Wojciechowski, Michał |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24578 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wiener-Pitt sets for compact Abelian groups Ohrysko, Przemysław Sanders, Tom Wojciechowski, Michał Functional Analysis Primary 43A10, Secondary 43A25 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}. |
| title | Wiener-Pitt sets for compact Abelian groups |
| topic | Functional Analysis Primary 43A10, Secondary 43A25 |
| url | https://arxiv.org/abs/2510.24578 |