FormalConjectures/GreensOpenProblems/82.lean

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Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL
Lean4 Domain Explanation Language

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

4 declarations, 3 contain sorry, 0 declared as axioms

Ben Green's Open Problem 82

References:

Imports (1)
  1. def Green82.minZeros The minimum number of zeros in $[0,1)$ of $\sum_{a \in A} \cos(2\pi a\theta)$ over all sets $A \subset \mathbb{Z}$ of size $n$ line 38 source

    Source documentation

    The minimum number of zeros in $[0,1)$ of $\sum_{a \in A} \cos(2\pi a\theta)$ over all sets $A \subset \mathbb{Z}$ of size $n$.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: PNat, ENat

    Proof uses: iInf, Subtype, Finset, Int, Eq, Nat, Finset.card, Set.ncard, Real, Set.ofPred, And, Membership.mem, Set, Set.Ico, OfNat.ofNat, Finset.sum, Real.cos, HMul.hMul, Real.pi

    25 instances, coercions and auxiliary constants left out.

    noncomputable def minZeros (n : ℕ+) : ℕ∞ :=
…
  2. theorem Green82.green_82 Let $A \subset \mathbb{Z}$ be a set of size $n$ sorry line 45 source

    Source documentation

    Let $A \subset \mathbb{Z}$ be a set of size $n$. For how many $\theta \in \mathbb{R}/\mathbb{Z}$ must we have $\sum_{a \in A} \cos(2\pi a\theta) = 0$? The answer is the function minZeros.

    depends on sorry what this means

    Statement uses: Eq, PNat, ENat, sorryAx, Bool.false, Green82.minZeros

    Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat

    1 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_82 : (answer(sorry) : ℕ+ → ℕ∞) = minZeros := by
…

    Mentions: Green82.minZeros

  3. theorem Green82.green_82_lower_bound Lower bound: every set has at least $(\log \log n)^{1-o(1)}$ zeros sorry line 50 source

    Source documentation

    Lower bound: every set has at least $(\log \log n)^{1-o(1)}$ zeros.

    depends on sorry what this means

    Statement uses: Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, Filter.Eventually, Finset, Int, Eq, Finset.card, LE.le, ENat, Nat.floor, HPow.hPow, Real.log, HSub.hSub, abs, Set.ncard and 8 more

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    66 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_82_lower_bound :
    ∃ o : ℕ → ℝ, o =o[atTop] (fun _ ↦ (1 : ℝ)) ∧
      ∀ᶠ n in atTop, ∀ A : Finset ℤ, A.card = n →
        (⌊(Real.log (Real.log n)) ^ (1 - |o n|)⌋₊ : ℕ∞) ≤
        ({θ : ℝ | θ ∈ Ico 0 1 ∧ ∑ a ∈ A, cos (2 * π * a * θ) = 0} : Set ℝ).ncard := by
…
  4. theorem Green82.green_82_upper_bound Upper bound: there exists $A$ with at most $C(n \log n)^{2/3}$ zeros sorry line 59 source

    Source documentation

    Upper bound: there exists $A$ with at most $C(n \log n)^{2/3}$ zeros.

    depends on sorry what this means

    Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, GE.ge, Finset, Int, Eq, Finset.card, LE.le, ENat, Set.ncard, Set.ofPred, Membership.mem, Set, Set.Ico, Finset.sum, Real.cos and 6 more

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    64 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_82_upper_bound :
    ∃ C > 0, ∀ n ≥ 1, ∃ A : Finset ℤ, A.card = n ∧
      ({θ : ℝ | θ ∈ Ico 0 1 ∧ ∑ a ∈ A, cos (2 * π * a * θ) = 0} : Set ℝ).ncard ≤
      (⌊C * (n * Real.log n) ^ (2 / 3 : ℝ)⌋₊ : ℕ∞) := by
…