Research Problems in Function Theory — Problem 5.65

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.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. The disc algebra $A(\overline{\mathbb D})$ consists of the functions continuous on $\overline{\mathbb D}$ and analytic in $\mathbb D$.

Statement recorded in the supplied source.

Does there exist a nonconstant $f\in A(\overline{\mathbb D})$ such that $f(e^{i\theta})\in f(\mathbb D)$ for Lebesgue-almost every $\theta\in[0,2\pi)$? The source update reports affirmative constructions, including one by Gol'dberg. [S2]

Short English statement

Can a nonconstant disc-algebra function take almost every boundary value also at an interior point? The original update reports affirmative constructions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305065 (AMR-022-5065). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 5.65, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.