Research Problems in Function Theory — Problem 7.47

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.

For compact $K\subset\mathbb C$, let $R(K)$ be the uniform closure on $K$ of rational functions with poles outside $K$. A point $x_0\in K$ is a peak point if some $h\in R(K)$ satisfies $h(x_0)=1$ and $|h(x)|<1$ for every $x\in K\setminus\{x_0\}$; a non-peak point is one for which no such $h$ exists. A curve terminating at $x_0$ means a continuous $\gamma:[0,1]\to K$ with $\gamma(1)=x_0$ and $\gamma(t)\ne x_0$ for $0\le t<1$.

Statement recorded in the supplied source.

For every compact $K\subset\mathbb C$ and every non-peak point $x_0$ for $R(K)$, does such a curve in $K$ terminating at $x_0$ exist? [S2] The source gives a sufficient Hausdorff-content condition for a straight segment, but also mentions examples where no such segment exists; the question asks for an arbitrary continuous curve. [S2]

Short English statement

Must every non-peak point for rational approximation be accessible through a continuous curve contained in the compact set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307047 (AMR-022-7047). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.47 and its complete update. https://arxiv.org/abs/1809.07200.