AI 中文总结
本文证明Gross定理中的例外集不必有对数容度零，构造了超越整函数及其正则局部逆，使其例外集含正容度紧集，且对任意s<1，该紧集有正s维豪斯多夫测度。
AI 中文摘要
1918年，Gross证明了平面内亚纯函数的正则局部逆可以沿从中心出发的每条射线解析延拓，除非方向属于勒贝格测度为零的集合。1977年，Nagasaka问及此例外集是否必须具有对数容度零。我们证明存在一个超越整函数及一个正则局部逆，使得其例外集包含一个具有正对数容度的紧集。事实上，对每个$0<s<1$，可选取这样的函数和逆，使得该紧集具有正的$s$维豪斯多夫测度。
英文摘要
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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