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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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FormalConjectures/ErdosProblems/

1048.lean

Retained formal statement6 of 8

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_1048.variants.pommerenke_diam_tendsto_zero9 linesExact file
∀ (r : ℝ),  1 < r →    ∀ (ε : ℝ),      0 < ε →        ∀ᶠ (n : ℕ) in Filter.atTop,          ∀ (f : Polynomial ℂ),            f = Polynomial.X ^ n - Polynomial.C (↑r ^ n) →              ∀ z ∈ Erdos1043.levelSet f,                Metric.ediam (connectedComponentIn (Erdos1043.levelSet f) z) < ENNReal.ofReal ε
SolvedStatement only, no proof