Research Problems in Function Theory — Problem 1.32

Definitions and mathematical statement

Let $f:\mathbb C\to\widehat{\mathbb C}$ be a nonconstant meromorphic function. Choose a finite regular value locally: take $z_0\in\mathbb C$ with $f(z_0)=w_0\in\mathbb C$ and $f'(z_0)\ne0$. The inverse function theorem gives a holomorphic inverse germ $g$ at $w_0$, with $g(w_0)=z_0$ and $f\circ g$ equal to the identity near $w_0$. For $\theta\in[0,2\pi)$ let $L_\theta(t)=w_0+te^{i\theta}$, $t\ge0$. Call the direction exceptional if the chosen germ cannot be analytically continued along every finite initial segment of $L_\theta$. This concerns continuation of that particular branch; it does not require one inverse branch on a neighbourhood of the union of all rays. Define \[ E(f,g,w_0)=\{\theta:\ g\text{ fails to continue along }L_\theta\}. \] Gross's star theorem states that $E(f,g,w_0)$ has one-dimensional Lebesgue measure zero. The problem is to determine a sharper universally valid description of this exceptional set, or to prove that a proposed strengthening is impossible. In particular, seek finer size restrictions or precise realizability results, with their dependence on the meromorphic function and inverse germ made explicit. No finite-order hypothesis is imposed.

Short English statement

How much smaller or more structured must the exceptional directions in Gross's inverse-continuation theorem be than an arbitrary set of angular measure zero?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2301032 (AMR-022-1032), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 1.32. Original source indexed by AMR-022-1032. https://arxiv.org/abs/1809.07200.