Research Problems in Function Theory — Problem 8.12

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 $\mathbb T=\partial\mathbb D$. Let $A$ be the boundary restrictions to $\mathbb T$ of the disc algebra: functions continuous on $\overline{\mathbb D}$ and analytic in $\mathbb D$. A sense-preserving circle homeomorphism is one preserving circular orientation. Define \[\widetilde A=\{f\circ\phi:f\in A,\ \phi:\mathbb T\to\mathbb T\text{ a sense-preserving homeomorphism}\},\quad q(e^{i\theta})=\sum_{n=1}^{\infty}2^{-n}e^{-i2^n\theta}.\] The series for $q$ is uniformly convergent.

Statement recorded in the supplied source.

Characterize membership in $\widetilde A$. Also, do there exist $h\in\widetilde A$ and a measurable $E\subset\mathbb T$ of positive arc length such that $h(\zeta)=q(\zeta)$ for every $\zeta\in E$? The source conjectures that the latter answer is negative. [S2]

Short English statement

Characterize circle reparametrizations of disc-algebra functions and decide whether one can agree with the stated lacunary series on a set of positive length.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308012 (AMR-022-8012). 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 8.12, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.