\(M\)-sets and measures (Q1190610)

A theorem of \textit{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).