Research Problems in Function Theory — Problem 6.110

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 a proper plane domain $U$, a Borel set $E\subseteq\partial U$, and $z\in U$, $\omega(z,E,U)$ denotes harmonic measure: the probability that planar Brownian motion begun at $z$ first exits $U$ through $E$. For an arbitrary set, use the associated outer measure; for sets off the boundary, use their intersection with $\partial U$. Let $n\geq1$, $0<\varepsilon<1/n$, and let $B_j=\{z:|z-x_j|\leq\varepsilon\}$, $x_j\in[0,1]$, be pairwise disjoint closed disks. Let $\Omega\subseteq\mathbb C$ be simply connected with $\bigcup_{j=1}^nB_j\subseteq\mathbb C\setminus\Omega$, and take $z_0\in\Omega$ with $\operatorname{dist}(z_0,B_j)\geq1$ for all $j$. Put $w(z_0,\Omega)=\omega(z_0,\partial\Omega\cap\bigcup_j B_j,\Omega)$.

Statement recorded in the supplied source.

For every $\delta>0$, does there exist $C_\delta<\infty$, depending only on $\delta$, such that for all the above configurations \[w(z_0,\Omega)\leq C_\delta(n\varepsilon)^{1/2-\delta}?\] [S2]

Short English statement

Does the harmonic measure of many small excluded disks obey the stated uniform near-square-root bound?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306110 (AMR-022-6110). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.110, complete problem and update. https://arxiv.org/abs/1809.07200.