Problem Statement
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 $\prod (z-z_i)$ with $z_i\in F$.
Is $\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\mu(F)=0$ whenever the transfinite diameter of $F$ is $\geq 1$?
Is $\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\mu(F)=0$ whenever the transfinite diameter of $F$ is $\geq 1$?
Categories:
Analysis
Progress
A problem of Erdős, Herzog, and Piranian [EHP58], who show that the answer is yes if $F$ is a line segment or disc, and that if the transfinite diameter is $<1$ then $\{ z: \lvert f(z)\rvert < 1\}$ always contains a disc of radius $\gg_F 1$.Erdős and Netanyahu [ErNe73] proved that if $F$ is also bounded and connected, with transfinite diameter $0<c<1$, then $\{ z: \lvert f(z)\rvert < 1\}$ always contains a disc of radius $\gg_c 1$.
The transfinite diameter of $F$, also known as the logarithmic capacity, is defined by\[\rho(F)=\lim_{n\to \infty}\sup_{z_1,\ldots,z_n\in F}\left(\prod_{i<j}\lvert z_i-z_j\rvert\right)^{1/\binom{n}{2}}.\]
Source: erdosproblems.com/1040 | Last verified: January 19, 2026