Research Problems in Function Theory — Problem 6.91

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 $\Omega\subset\mathbb C$ be a domain, choose $p\in\Omega$, and let $\omega_\Omega^p$ denote harmonic measure on its finite boundary (the distribution of the first finite boundary hit by planar Brownian motion started at $p$). A set $E\subset\partial\Omega$ has full harmonic measure if $\omega_\Omega^p(\partial\Omega\setminus E)=0$. For an arc $C_r$ on the circle $|w-z|=r$, its angular measure is $\theta(C_r)=\operatorname{length}(C_r)/r$, in radians. The integral below requires the selected angular measures to be measurable in $r$.

Statement recorded in the supplied source.

Does every $\Omega$ have a full-harmonic-measure set $E$ such that for each $z\in E$ one can choose circular arcs $C_r\subset\Omega\cap\{w:|w-z|=r\}$ for all sufficiently small $r>0$, with \[\lim_{R\to0^+}\frac1{\pi R^2}\int_0^R\theta(C_r)r\,dr=\frac12?\] [S2] The source records McMillan's result for simply connected domains. Its separate weak sequence statement is not substituted for this averaged target. [S2]

Short English statement

Does almost every boundary point for harmonic measure admit small interior circular arcs with the specified averaged half-disc angular density?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306091 (AMR-022-6091). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 6, Problem 6.91 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.