1038.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 1038

Reference:

erdosproblems.com/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)

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

theorem Erdos1038.erdos_1038.parts.i 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]? sorry line 39 source
Source documentation
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]?

depends on sorry what this means
Statement uses: Nat, Eq, ENNReal, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, iInf, Subtype, Polynomial, Real, And, Polynomial.Monic, Ne, Multiset.card, Multiset.filter, Membership.mem and 11 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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} := by
…
```
theorem Erdos1038.erdos_1038.parts.ii 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) sorry line 49 source
Source documentation
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].

depends on sorry what this means
Statement uses: Nat, Eq, ENNReal, HMul.hMul, OfNat.ofNat, HPow.hPow, Real, HDiv.hDiv, iSup, Subtype, Polynomial, And, Polynomial.Monic, Multiset.card, Multiset.filter, Membership.mem, Set, Set.Icc, Neg.neg, Polynomial.roots and 7 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
82 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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} := by
…
```
theorem Erdos1038.erdos_1038.variants.inf_upperBound 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 sorry line 58 source
Source documentation
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.

depends on sorry what this means
Statement uses: Nat, LT.lt, ENNReal, iInf, Subtype, Polynomial, Real, And, Polynomial.Monic, Ne, OfNat.ofNat, Eq, Multiset.card, Multiset.filter, Membership.mem, Set, Set.Icc, Neg.neg, Polynomial.roots, Polynomial.natDegree and 7 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
63 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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 := by
…
```
theorem Erdos1038.erdos_1038.varaints.inf_lowerBound 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 sorry line 66 source
Source documentation
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.

depends on sorry what this means
Statement uses: Nat, LE.le, ENNReal, HSub.hSub, HPow.hPow, Real, OfNat.ofNat, HDiv.hDiv, iInf, Subtype, Polynomial, And, Polynomial.Monic, Ne, Eq, Multiset.card, Multiset.filter, Membership.mem, Set, Set.Icc and 9 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
86 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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} := by
…
```