Problem 1040

Erdős Problem #1040: declared status 'open'. Formalized: no. 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 $\geq 1$? Prize: no. Tags: analysis.