Wiener-Pitt sets for compact Abelian groups

Journal article

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 (an)∞ n=1 ∈ c0(N), for every strongly continuous µ ∈ M(G) with ∥µ∥ ≤ 1 and µb(Gb) ⊂ {an : 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 the article ‘On the relationships between Fourier-Stieltjes coefficients and spectra of measures’ published in 2014.

Authors: Ohrysko, P; Sanders, T; Wojciechowski, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Journal of Fourier Analysis and Applications
• Volume: 32
• Issue: 2
• Article number: 30
• Publication date: 2026-03-04
• Acceptance date: 2026-01-04
• DOI: 10.1007/s00041-026-10239-1
• EISSN: 1531-5851
• ISSN: 1069-5869
• Language: English
• Keywords: natural spectrum; Wiener–Pitt phenomenon; measure algebras