FormalConjectures/GreensOpenProblems/9.lean

GitHub source file

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

Green's Open Problem 9

References:

Imports (1)
Imported by 1 file
  1. def Green9.r The quantity $r_k(N)$, defined as the size of the largest subset of $\{1, \dots, N\}$ without non-trivial $k$-term arithmetic progressions line 39 source

    Source documentation

    The quantity $r_k(N)$, defined as the size of the largest subset of $\{1, \dots, N\}$ without non-trivial $k$-term arithmetic progressions.

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

    Statement uses: Nat

    Proof uses: Finset.maxAPFreeCard, Finset.Icc, OfNat.ofNat

    4 instances, coercions and auxiliary constants left out.

    noncomputable def r (k N : ℕ) : ℕ := (Finset.Icc 1 N).maxAPFreeCard k
…
  2. theorem Green9.green_9_i Problem 9 (i): is $r_3(N) \ll N(\log N)^{-10}$? sorry line 47 source

    Source documentation

    Problem 9 (i): is $r_3(N) \ll N(\log N)^{-10}$?

    Solved in [BlSi20].

    depends on sorry what this means

    Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Green9.r, OfNat.ofNat, HMul.hMul, HPow.hPow, Real.log, Neg.neg

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

    26 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem green_9_i :
    (fun (N : ℕ) ↦ (r 3 N : ℝ)) ≪ fun (N : ℕ) ↦ (N : ℝ) * (Real.log N) ^ (-10 : ℝ) := by
…

    Mentions: Green9.r

  3. theorem Green9.green_9_ii Problem 9 (ii): is $r_5(N) \ll N(\log N)^{-c}$? sorry line 55 source

    Source documentation

    Problem 9 (ii): is $r_5(N) \ll N(\log N)^{-c}$?

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, Green9.r, HMul.hMul, HPow.hPow, Real.log, Neg.neg

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

    26 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem green_9_ii : answer(sorry) ↔
    ∃ c > (0 : ℝ), (fun (N : ℕ) ↦ (r 5 N : ℝ))
      ≪ fun (N : ℕ) ↦ (N : ℝ) * (Real.log N) ^ (-c) := by
…

    Mentions: c, Green9.r

  4. theorem Green9.green_9_iii Problem 9 (iii): is $r_4(\mathbf{F}_5^n) \ll N^{1-c}$, where $N=5^n$? sorry line 64 source

    Source documentation

    Problem 9 (iii): is $r_4(\mathbf{F}_5^n) \ll N^{1-c}$, where $N=5^n$?

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, Finset.maxAPFreeCard, 𝔽₅, Fin, ZMod, Finset.univ, HPow.hPow, HSub.hSub

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

    58 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem green_9_iii : answer(sorry) ↔
    ∃ c > (0 : ℝ), (fun (n : ℕ) ↦ ((Finset.univ : Finset (𝔽₅ n)).maxAPFreeCard 4 : ℝ))
      ≪ fun (n : ℕ) ↦ ((5 : ℝ) ^ n) ^ (1 - c) := by
…

    Mentions: c, 𝔽₅