Research Problems in Function Theory — Problem 5.20

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. Planar measure means two-dimensional Lebesgue measure on the value plane $\mathbb C$; almost every boundary point refers to arc-length measure on $\mathbb T$.

Statement recorded in the supplied source.

For every holomorphic $f$ on $\mathbb D$, is it true that at almost every $\zeta\in\mathbb T$, either $f$ has a finite angular limit at $\zeta$, or there is a set $E_\zeta\subset\mathbb C$ of planar measure zero such that, for every $a\in\mathbb C\setminus E_\zeta$ and every Stolz angle at $\zeta$, there is a sequence of distinct points $z_j$ in that angle tending to $\zeta$ with $f(z_j)=a$? More generally, what strengthening of a merely dense set of infinitely assumed values is valid in the second alternative? [S2]

Short English statement

Can the dense set of repeatedly assumed values in Plessner’s boundary alternative be enlarged to all values outside a planar null set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305020 (AMR-022-5020). 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 5.20, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.