Non-tangential ranges of holomorphic functions at Plessner points

Abstract

Consider the random lacunary series $f(z) = \sum_{k=1}^\infty \frac{\xi_k}{\sqrt{k}} \, z^{2^k}$ on the unit disk, where $\{ \xi_k \}$ are independent standard complex Gaussian random variables. We show that almost surely, a.e. $\zeta \in \partial \mathbb{D}$ is a Plessner point of $f$, yet the image of every Stolz angle with vertex at $\zeta$ has asymptotic density zero. This gives a negative answer to questions of Collingwood and Baernstein concerning possible strengthenings of Plessner's theorem. In this example, for a.e. $\zeta \in \partial \mathbb{D}$, the non-tangential range of $f$ at $\zeta$ has zero area. The non-tangential range cannot be much smaller: we show that for an arbitrary holomorphic function on the unit disk, the non-tangential range has Hausdorff dimension 2 at almost every Plessner point.