A converse to the Lusin-Privalov radial uniqueness theorem

Abstract:

Let $E$ be a subset of the unit circumference $C$. If for every nonempty open arc $A$ of $C$, the set $E$ is not both metrically dense and of second category in $A$, then there exists a nonconstant analytic function $f$ on the open unit disk $\Delta$, such that ${f^ * }(\eta ) = 0$, $\eta \in E$, where ${f^ * }$ is the radial limit function of $f$.