Research Problems in Function Theory — Problem 7.18

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 $\Gamma\subset\mathbb C$ be a Jordan curve, meaning the image of a continuous injective parametrisation of the circle, with $0$ in its bounded complementary component. Write $C(\Gamma)$ for the complex continuous functions with the uniform norm. Since $0\notin\Gamma$, the functions $z\mapsto z^n$ are continuous on $\Gamma$ for every $n\in\mathbb Z$. A family spans $C(\Gamma)$ when its finite complex linear combinations are dense in this norm.

Statement recorded in the supplied source.

Characterise the curves $\Gamma$ for which there exists an infinite set $E\subset\mathbb Z$ such that \[\overline{\operatorname{span}\{z^n|_\Gamma:n\in\mathbb Z\setminus E\}}^{\|\cdot\|_\infty}=C(\Gamma).\] In other words, under what conditions on $\Gamma$ can infinitely many Laurent powers be omitted without losing density? [S2] The original notes Wermer's density result after deleting the constant power for curves of infinite length, and related results for finitely many deleted powers. These are context for the infinite-deletion question. [S2]

Short English statement

Determine which Jordan curves permit deleting infinitely many integer powers from a uniformly dense Laurent-power family.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307018 (AMR-022-7018). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.18 and its complete update. https://arxiv.org/abs/1809.07200.