A converse to the Lusin-Privalov radial uniqueness theorem

Abstract

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