FormalConjectures/ErdosProblems/996.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 996

Reference:

Imports (1)
Imported by 1 file
  1. def Erdos996.fourierPartial line 37 source

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

    Statement uses: Real, Fact, LT.lt, OfNat.ofNat, Subtype, MeasureTheory.AEEqFun, AddCircle, Complex, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Membership.mem, AddSubgroup, MeasureTheory.Lp, ENNReal, Nat

    Proof uses: Finset.sum, Int, Finset.Icc, Neg.neg, HSMul.hSMul, fourierCoeff, ContinuousMap, fourier

    76 instances, coercions and auxiliary constants left out.

  2. theorem Erdos996.erdos_996 Documented declaration: erdos_996 sorry line 45 source

    Source documentation

    Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Subtype, MeasureTheory.AEEqFun, AddCircle, Complex, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Membership.mem, AddSubgroup, MeasureTheory.Lp, ENNReal, Nat, IsLacunary, Asymptotics.IsBigO and 18 more

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

    170 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_996 : answer(sorry) ↔
    ∃ (C : ℝ), 0 < C ∧ ∀ (f : Lp ℂ 2 (haarAddCircle (T := 1))) (n : ℕ → ℕ),
…
  3. theorem Erdos996.erdos_996.variants.log2 The following theorem is proved in [Ma66] sorry line 57 source

    Source documentation

    The following theorem is proved in [Ma66].

    depends on sorry what this means

    Statement uses: Real, LT.lt, OfScientific.ofScientific, Bool.true, Subtype, MeasureTheory.AEEqFun, AddCircle, OfNat.ofNat, Complex, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Membership.mem, AddSubgroup, MeasureTheory.Lp, ENNReal, Nat, IsLacunary, Asymptotics.IsBigO, Filter.atTop, ENNReal.toReal and 16 more

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

    172 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_996.variants.log2 : ∀ (C : ℝ), 0.5 < C →
    ∀ (f : Lp ℂ 2 (haarAddCircle (T := 1))) (n : ℕ → ℕ),
…