On the zero sets of the Fourier transform of singular measures (Q1714985)

The author shows that the class of sets \(E\) with the property that there exists a measure singular with respect to the Lebesgue measure whose Fourier transform tends to 0 at infinity and vanishes on \(E\) contains the Helson sets. He calls \(E \subset \mathbb R\) a \(\theta\)-set iff for every \(g\in C0(E)\) there exists a singular measure \(\mu\in M(R)\) such that \(\mu(t) = g(t)\) for \(t\in E\). The main result is: The class of Helson sets coincides with the class of \(\theta\)-sets. He shows that the class of sets \(E\) with the property that there exists a measure singular with respect to the Lebesgue measure whose Fourier transform tends to 0 at infinity and vanishes on \(E\) contains the Helson sets.