On the zeros of functions with finite Dirichlet integral (Q1916064)

Let \({\mathcal D}\) be the Dirichlet space of functions analytic in \(\Delta = \{z : |z |< 1\}\) and having finite Dirichlet integral \[ \int_\Delta |f' (z) |^2 dx dy. \] It is known that the zero-set of any function of \({\mathcal D}\) satisfies the Blaschke condition but the converse is not true. The author calls \(A \subset \Delta\) a Blaschke set for \({\mathcal D}\) if any subsequence of \(A\) satisfying the Blaschke condition is the zero-set of a function of \({\mathcal D}\). The main result is the following Theorem. A set \(A \subset \Delta\) is Blaschke for \({\mathcal D}\) if and only if \[ - \int^\pi_{- \pi} \log \text{dist} (A, e^{it}) dt < \infty. \tag{1} \] Moreover, if (1) holds, then there is an outer function \(F \in {\mathcal D}\) with the property that \(BF \in {\mathcal D}\) for every Blaschke product \(B\) with zeros in \(A\).