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import FormalConjecturesUtil
import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegreeErdős Problem 1038
[Tao25] Tao, Terence. Sublevel Sets of Logarithmic Potentials. Terry Tao’s Blog, Dec. 2025 (https://terrytao.wordpress.com/wp-content/uploads/2025/12/erdos-1038-1.pdf)
open scoped Real ENNRealopen MeasureTheory
namespace Erdos1038
What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such
that all of its roots are real and contained in [-1,1]?
@[category research open, AMS 28]
theorem erdos_1038.parts.i (n : ℕ) : answer(sorry) =
⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧
(f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree},
volume {x | |f.1.eval x| < 1} := n:ℕ⊢ sorry = ⨅ f, volume {x | |Polynomial.eval x ↑f| < 1}
All goals completed! 🐙
The supremum of |{x ∈ ℝ : |f x| < 1}| over all monic polynomials f such that
all of its roots are real and contained in [-1,1] is 2 * 2 ^ (1 / 2). This is proved in
[Tao25].
@[category research solved, AMS 28]
theorem erdos_1038.parts.ii (n : ℕ) : 2 * 2 ^ (1 / 2 : ℝ) =
⨆ f : {f : Polynomial ℝ // f.Monic ∧
(f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree},
volume {x | |f.1.eval x| < 1} := n:ℕ⊢ 2 * 2 ^ (1 / 2) = ⨆ f, volume {x | |Polynomial.eval x ↑f| < 1}
All goals completed! 🐙
The infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that
all of its roots are real and contained in [-1,1] is < 1.835.
@[category research solved, AMS 28]
theorem erdos_1038.variants.inf_upperBound (n : ℕ) : ⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧
(f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree},
volume {x | |f.1.eval x| < 1} < 1.835 := n:ℕ⊢ ⨅ f, volume {x | |Polynomial.eval x ↑f| < 1} < 1.835
All goals completed! 🐙
The infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that
all of its roots are real and contained in [-1,1] is ≥ 2 ^ (4 / 3) - 1.
@[category research solved, AMS 28]
theorem erdos_1038.varaints.inf_lowerBound (n : ℕ) : 2 ^ (4 / 3 : ℝ) - 1 ≤
⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧
(f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree},
volume {x | |f.1.eval x| < 1} := n:ℕ⊢ 2 ^ (4 / 3) - 1 ≤ ⨅ f, volume {x | |Polynomial.eval x ↑f| < 1}
All goals completed! 🐙
end Erdos1038