The descriptive complexity of Helson sets (Q1913497)

A closed subset \(E\) of the circle group \(T\) is called a Helson set if every continuous complex-valued function on \(E\) can be extended to a function on \(T\) with absolutely convergent Fourier series. We denote by \({\mathcal H}\) the class of Helson subsets of \(T\). We are interested in the descriptive properties of \({\mathcal H}\). We shall need the following definitions: a subset of a compact metric space is called a \(G_{\delta \sigma}\) set if it is the union of countably many \(G_\delta\) sets and an \(F_{\sigma \delta}\) set if its complement is \(G_{\delta \sigma}\). In the sequel we follow the notations of \textit{Y. N. Moschovakis} [Descriptive set theory, Amsterdam (1980; Zbl 0433.03025)]. Thus, the symbols \(\Pi^0_2\), \(\Sigma^0_3\), \(\Pi^0_3\) respectively mean \(G_\delta\), \(G_{\delta \sigma}\), \(F_{\sigma \delta}\). However, we sometimes use \(G_\delta\) instead of \(\Pi^0_2\). Let \({\mathcal K} (T)\) be the space of all compact subsets of \(T\) equipped with its (metric, compact) Hausdorff topology. One natural question (at least for some people) is to find the exact Borel class of \({\mathcal H}\) as a subset of \({\mathcal K} (T)\) (this is what ``descriptive properties'' meant). It is easy to check (Section 1) that \({\mathcal H}\) is \(\Sigma^0_3\). We show that \({\mathcal H}\) is a true \(\Sigma^0_3\) set (that is, \(\Sigma^0_3\) but not \(\Pi^0_3)\). We do this in two ways. First (Section 2) we prove that even inside the countable sets \({\mathcal H}\) is true \(\Sigma^0_3\). Then (Section 3) we get the same conclusion for perfect Helson sets. In fact, our result is slightly more general: we show that for any \(M_0\) set \(E\), the perfect Helson sets contained in \(E\) form a true \(\Sigma^0_3\) subset of \({\mathcal K} (E) = \{F \in {\mathcal K} (T); \;F \subseteq E\}\) (the definition of an \(M_0\) set will be given in Section 3). The proof also yields that some other natural classes of thin sets, like the \(WTP\), \(U'\) or \(U_0'\) sets, are true \(\Sigma^0_3\) within any \(M_0\) set.