Erdős Problem #1040 — Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of $\lvert \{ z: \lvert f(z)\rvert < 1\}\rvert,$ as $f$ ranges over all polynomials of the shape $\prod (z-z_i)$…

Erdős, Herzog, and Piranian posed the question in 1958, asking whether $\mu(F)$ is determined solely by the transfinite diameter of $F$. The answer is now negative.

Known results

• Erdős, Herzog, and Piranian (1958): affirmative for line segments and discs; they also obtained a positive lower-radius bound when the transfinite diameter is less than $1$.
• Erdős and Netanyahu (1973): for bounded connected $F$ with transfinite diameter $c\in(0,1)$, every relevant sublevel set contains a disc of radius bounded below in terms of $c$.

April 2026 disproof and extensions

“A note on the Erdős minimal area problem” constructs compact sets $K_1,K_2$ with equal capacity but unequal $\mu$, even allowing arbitrarily different values at fixed capacity. It also proves $\mu(K)=0$ for every compact $K$ with capacity greater than $1$; the general capacity-one case remains open.

Current status (as of June 2026): The original transfinite-diameter question is settled negatively; the general capacity-one vanishing problem remains open.