Wiener-Pitt sets for compact Abelian groups

2025/10/28 by Ohrysko, Przemysław, Sanders, Tom, Wojciechowski, Michał
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 43A10 #Secondary 43A25

paper · doi:10.48550/arxiv.2510.24578

Abstract

Suppose that G is a compact Hausdorff Abelian group. We say μ∈ M(G) is strongly continuous if |μ|(x+H)=0 for any x ∈ G and any H ≤ G that is closed and of infinite index. We prove that for any sufficiently rapidly decreasing sequence (a_n)_n=1^∞∈ c₀(ℕ), for every strongly continuous μ∈ M(G) with ‖μ‖ ≤ 1 and \widehatμ(\widehatG)⊂ \a_n: n ∈ ℕ\∪\0\, the measure μ∗μ 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 \citeow.

Citations

• On a Conjecture of Littlewood and Idempotent Measures

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