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Erdős problem 1048

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If f∈C[x]f\in \mathbb{C}[x] is a monic polynomial with all roots satisfying ∣z∣≤r\lvert z\rvert \leq r for some r<2r<2, then must {z:∣f(z)∣<1}\{ z: \lvert f(z)\rvert <1\} have a connected component with diameter >2−r>2-r?

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1048.lean

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If f∈C[x]f\in \mathbb{C}[x] is a monic polynomial with all roots satisfying ∣z∣≤r\lvert z\rvert \leq r for some r<2r<2, then must {z:∣f(z)∣<1}\{ z: \lvert f(z)\rvert <1\} have a connected component with diameter >2−r>2-r?

A problem of Erdős, Herzog, and Piranian [EHP58].

Pommerenke [Po61] proved the answer is no for r>1r>1, showing that if f(z)=zn−rnf(z)=z^n-r^n then {z:∣f(z)∣≤1}\{ z: \lvert f(z)\rvert \leq 1\} has nn connected components, all with diameter →0\to 0 as n→∞n\to \infty.

FormalConjectures/ErdosProblems/1048.leanErdos1048.erdos_10488 linesExact file
False ↔  ∀ (r : ℝ) (f : Polynomial ℂ),    r < 2 →      f.Monic →        f.degree ≥ 1 →          (∀ z ∈ f.roots, ‖z‖ ≤ r) →            ∃ z ∈ Erdos1048.openLevelSet f,              ENNReal.ofReal (2 - r) < Metric.ediam (connectedComponentIn (Erdos1048.openLevelSet f) z)
SolvedProof has a holelean4external proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.