Erdős Problem #1048
DISPROVED LEANdatabase updated 2026-01-27
The statement as posed is false. A solution has also been checked by machine, in Lean.
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The set \(\{ z \in \mathbb{C} : \lvert f(z)\rvert < 1\}\)
If \(f\in \mathbb{C}[x]\) is a monic polynomial with all roots satisfying \(\lvert z\rvert \leq r\) for some \(0 < r < 2\), then must \[\{ z: \lvert f(z)\rvert <1\}\] have a connected component with diameter \(>2-r\)? A problem of Erdős, Herzog, and Piranian [EHP58]. Pommerenke [Po61] proved the answer is no for \(r>1\), showing that if \(f(z)=z^n-r^n\) then \(\{ z: \lvert f(z)\rvert \leq 1\}\) has \(n\) connected components, all with diameter \(\to 0\) as \(n\to \infty\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 7 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1048.Prizenone
Topicsanalysis
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1048
Attempts
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