Erdős Problem #1040

The problem

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 $\geq 1$?

Statement quoted from erdosproblems.com/1040, maintained by Thomas Bloom. Original source: [EHP58,p.135]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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