Research Problems in Function Theory — Problem 9.12

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 $\Gamma$ be the boundary of a square, scaled so its logarithmic capacity is $1$. For a compact planar set $K$, capacity is \[\operatorname{cap}(K)=\lim_{n\to\infty}\max_{z_1,\ldots,z_n\in K}\left(\prod_{1\le i<j\le n}|z_i-z_j|\right)^{2/(n(n-1))}.\] An $n$-point Fekete configuration on $\Gamma$ maximizes this product; take one for each $n\ge2$ and put $U_n(z)=n^{-1}\sum_{j=1}^n\log|z-z_j|$. Let $\phi$ map the unbounded component of $\widehat{\mathbb C}\setminus\Gamma$ conformally onto $\{|w|>1\}$, fixing infinity. Define $U(z)=\log|\phi(z)|$ outside the square and $U(z)=0$ inside.

Statement recorded in the supplied source.

Prove the estimate $U_n(z)-U(z)=O(n^{-1})$ away from $\Gamma$: for every compact $L\subset\mathbb C\setminus\Gamma$, show that there is a constant $C_L$ with \[\sup_{z\in L}|U_n(z)-U(z)|\le C_L/n\quad(n\ge2).\] [S2]

Short English statement

Prove first-order convergence of Fekete-point logarithmic potentials to the equilibrium potential for a capacity-one square, uniformly away from its boundary.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309012 (AMR-022-9012). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 9.12, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.