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import FormalConjecturesUtil
import FormalConjectures.ErdosProblems.«1041»Erdős Problem 1044
References:
[EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
[Ta26] Tang, Quanyu, On Erdős Problem 1044 (2026), github.com/QuanyuTang/erdos-problem-1044.
open Polynomial ENNRealnamespace Erdos1044The polynomials under consideration: those of the form $f(z)=\prod_{i=1}^n(z-z_i)$, for some $n\geq 1$, where $\lvert z_i\rvert\leq 1$ for all $i$.
def IsAdmissible (f : ℂ[X]) : Prop :=
∃ n : ℕ, 0 < n ∧ ∃ z : Fin n → ℂ, (∀ i, ‖z i‖ ≤ 1) ∧ f = ∏ i, (X - C (z i))$\Lambda(f)$, the maximum of the lengths of the boundaries of the connected components of ${ z: \lvert f(z)\rvert<1}$, where the length of a subset of $\mathbb{C}$ is its $1$-dimensional Hausdorff measure.
noncomputable def maxBoundaryLength (f : ℂ[X]) : ℝ≥0∞ :=
⨆ z ∈ {w : ℂ | ‖f.eval w‖ < 1},
Erdos1041.length (frontier (connectedComponentIn {w : ℂ | ‖f.eval w‖ < 1} z))Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ where $\lvert z_i\rvert\leq 1$ for all $i$. If $\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of $$ { z: \lvert f(z)\rvert<1} $$ then determine the infimum of $\Lambda(f)$.
A problem of Erdős, Herzog, and Piranian [EHP58].
This has been resolved by Tang, who proved that the infimum of $\Lambda(f)$ over all such $f$ is $2$.
@[category research solved, AMS 30, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1044.lean"]
theorem erdos_1044 :
IsGLB {L : ℝ≥0∞ | ∃ f : ℂ[X], IsAdmissible f ∧ maxBoundaryLength f = L} answer(2) := ⊢ IsGLB {L | ∃ f, IsAdmissible f ∧ maxBoundaryLength f = L} 2
All goals completed! 🐙This has been resolved by Tang, who proved that the infimum of $\Lambda(f)$ over all such $f$ is $2$.
@[category research solved, AMS 30, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1044.lean"]
theorem erdos_1044.variants.infimum_eq_two :
IsGLB {L : ℝ≥0∞ | ∃ f : ℂ[X], IsAdmissible f ∧ maxBoundaryLength f = L} 2 := ⊢ IsGLB {L | ∃ f, IsAdmissible f ∧ maxBoundaryLength f = L} 2
All goals completed! 🐙Tang also suggests that, if the degree $n$ is fixed, then the infimum over all such $f$ of degree $n$ is attained by $f_n(z)=z^n-1$.
@[category research open, AMS 30]
theorem erdos_1044.variants.fixed_degree (n : ℕ) (hn : 0 < n) :
IsLeast {L : ℝ≥0∞ | ∃ f : ℂ[X], IsAdmissible f ∧ f.natDegree = n ∧ maxBoundaryLength f = L}
(maxBoundaryLength (X ^ n - 1)) := n:ℕhn:0 < n⊢ IsLeast {L | ∃ f, IsAdmissible f ∧ f.natDegree = n ∧ maxBoundaryLength f = L} (maxBoundaryLength (X ^ n - 1))
All goals completed! 🐙Tang also suggests that, if the degree $n$ is fixed, then the infimum over all such $f$ of degree $n$ is attained by $f_n(z)=z^n-1$ (and proves this for $n=1$ and $n=2$).
@[category research solved, AMS 30]
theorem erdos_1044.variants.fixed_degree_of_le_two (n : ℕ) (hn : n = 1 ∨ n = 2) :
IsLeast {L : ℝ≥0∞ | ∃ f : ℂ[X], IsAdmissible f ∧ f.natDegree = n ∧ maxBoundaryLength f = L}
(maxBoundaryLength (X ^ n - 1)) := n:ℕhn:n = 1 ∨ n = 2⊢ IsLeast {L | ∃ f, IsAdmissible f ∧ f.natDegree = n ∧ maxBoundaryLength f = L} (maxBoundaryLength (X ^ n - 1))
All goals completed! 🐙end Erdos1044