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\(M\)-sets and measures. (English) Zbl 0793.42006
A theorem of R. Lyons [Ann. Math., II. Ser. 122, 155-170 (1985; Zbl 0583.43006)] states that a measure on the circle group has Fourier- Stieltjes coefficients not tending to 0 iff it has positive mass on some (closed) \(U_ 0\)-set (set of uniqueness in the wide sense). The author answers the outstanding question whether the above theorem can be restated with \(U\)-sets (sets of uniqueness) instead of \(U_ 0\)-sets in the negative by constructing a measure \(\mu\) whose Fourier-Stieltjes coefficients do not tend to 0, but still vanishes on all sets of uniqueness, i.e., the only closed sets of positive \(\mu\)-measure are \(M\)- sets (sets of multiplicity).

MSC:
42A63 Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A55 Lacunary series of trigonometric and other functions; Riesz products
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