Erdős Problem #1048

The problem

If $f\in \mathbb{C}[x]$ is a monic polynomial with all roots satisfying $\lvert z\rvert \leq r$ for some $r<2$, then must

$$\{ z: \lvert f(z)\rvert <1\}$$

have a connected component with diameter $>2-r$?

Statement quoted from erdosproblems.com/1048, maintained by Thomas Bloom. Original source: [EHP58,p.142]. Current status there: disproved (Lean).

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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