Erdős Problem 1040

### What is the conjecture

https://www.erdosproblems.com/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 $\prod (z-z_i)$ 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 $\geq 1$?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

Read the linked Erdős Problems page to understand the conjecture, then inspect the repository for existing formalized Erdős problems and their organization. The work is complete when this conjecture is represented as a compiling formal statement in the repository and its meaning matches the source conjecture.

Written by the indexing model from the issue text.