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Non-tangential ranges of holomorphic functions at Plessner points
Abstract
Consider the random lacunary series f(z) = ∑k=1∞ ξk/√k z2k on the unit disk, where { ξk } are independent standard complex Gaussian random variables. We show that almost surely, a.e. ζ∈ ∂ 𝔻 is a Plessner point of f, yet the image of every Stolz angle with vertex at ζ 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. ζ∈ ∂ 𝔻, the non-tangential range of f at ζ 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.
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BibTeX
@misc{ivrii2026nontangential,
title = {{Non-tangential ranges of holomorphic functions at Plessner points}},
author = {Oleg Ivrii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18785},
eprint = {2609.18785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18785},
}
RIS
TY - UNPB
TI - Non-tangential ranges of holomorphic functions at Plessner points
AU - Oleg Ivrii
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18785
UR - https://arxiv.org/abs/2609.18785
AB - Consider the random lacunary series f(z) = ∑_k=1^∞ ξ_k/√k z^2^k on the unit disk, where { ξ_k } are independent standard complex Gaussian random variables. We show that almost surely, a.e. ζ∈ ∂ 𝔻 is a Plessner point of f, yet the image of every Stolz angle with vertex at ζ 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. ζ∈ ∂ 𝔻, the non-tangential range of f at ζ 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.
ER -