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Exceptional Sets of Positive Capacity in Gross's Star Theorem
Abstract
In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. In 1977, Nagasaka asked whether this exceptional set must have logarithmic capacity zero. We prove that there exist a transcendental entire function and a regular local inverse for which the exceptional set contains a compact set of positive logarithmic capacity. In fact, for every 0<s<1, such a function and inverse can be chosen so that this compact set has positive s-dimensional Hausdorff measure.
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BibTeX
@misc{nadi2026exceptional,
title = {{Exceptional Sets of Positive Capacity in Gross's Star Theorem}},
author = {Sina Nadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29797},
eprint = {2609.29797},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29797},
}
RIS
TY - UNPB
TI - Exceptional Sets of Positive Capacity in Gross's Star Theorem
AU - Sina Nadi
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29797
UR - https://arxiv.org/abs/2609.29797
AB - In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. In 1977, Nagasaka asked whether this exceptional set must have logarithmic capacity zero. We prove that there exist a transcendental entire function and a regular local inverse for which the exceptional set contains a compact set of positive logarithmic capacity. In fact, for every 0<s<1, such a function and inverse can be chosen so that this compact set has positive s-dimensional Hausdorff measure.
ER -