Research Problems in Function Theory — Problem 3.23

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. The positive Green kernel of $\mathbb D$ is \[G_{\mathbb D}(z,a)=\log\left|\frac{1-\overline a z}{z-a}\right|\quad(z\ne a),\qquad G_{\mathbb D}(a,a)=+\infty. \] A Green potential is $u(z)=\int_{\mathbb D}G_{\mathbb D}(z,a)\,d\mu(a)$ for a positive Borel measure $\mu$ for which the potential is well defined. Assume $u(0)=1$, and for $r\ge0$ let $E_r(u)=\{z\in\mathbb D:u(z)>r\}$. Write $D(a,s)=\{z:|z-a|<s\}$ for a disc of radius $s>0$.

Statement recorded in the supplied source.

Does there exist one function $g:[0,\infty)\to[0,\infty)$ with $g(r)\to0$ as $r\to\infty$ such that, for every normalized Green potential $u$ and every $r\ge0$, there is a finite or countable covering \[E_r(u)\subseteq\bigcup_kD(a_k,s_k),\qquad\sum_ks_k\le g(r)?\] Ask the same question with $u$ replaced by any positive harmonic function on $\mathbb D$ normalized by $u(0)=1$. The covering may depend on both $u$ and $r$, but the function $g$ must be universal for the indicated class. [S2]

Short English statement

Can high-value sets of every normalized Green potential be covered by discs whose total radii tend uniformly to zero? Ask the same for positive harmonic functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303023 (AMR-022-3023). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 3.23, its complete Update 3.23, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.