Research Problems in Function Theory — Problem 5.29

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$. For $\zeta\in\mathbb T$, a Stolz angle means a nondegenerate Euclidean angular sector with vertex $\zeta$, whose two rays enter $\mathbb D$ nontangentially, restricted to a sufficiently small neighborhood of $\zeta$. A finite angular limit is a common finite limit as $z\to\zeta$ within every such sector.A $G_\delta$ subset of $\mathbb T$ is a countable intersection of relatively open subsets. A Fatou point of a bounded holomorphic function is a boundary point where its finite angular limit exists. The boundary value $f^*(\zeta)$ denotes that angular limit.

Statement recorded in the supplied source.

For every $G_\delta$ set $E\subset\mathbb T$ of arc-length measure zero, does there exist a bounded holomorphic function $f\not\equiv0$ on $\mathbb D$ such that every point of $\mathbb T$ is a Fatou point and $f^*(\zeta)=0$ for every $\zeta\in E$? [S2]

Short English statement

Historical question: can a nonzero bounded holomorphic function vanish on any prescribed null $G_\delta$ boundary set while having an angular limit at every boundary point? The source reports an affirmative solution.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305029 (AMR-022-5029). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.29, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.