FormalConjectures/ErdosProblems/818.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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 818

References:

Imports (1)
  1. theorem Erdos818.erdos_818 Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$ sorry line 45 source

    Source documentation

    Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$. Is it true that $$\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{(\log \lvert A\rvert)^C}$$ for some constant $C>0$?

    This was proved by Solymosi [So09d], in the strong form $$\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{\log \lvert A\rvert}.$$ See also [52].

    depends on sorry what this means

    Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, And, Finset, Int, LE.le, Nat, Finset.card, HAdd.hAdd, HMul.hMul, HDiv.hDiv, HPow.hPow, Real.log

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

    48 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_818 : answer(True) ↔
    ∀ K : ℝ, 0 < K → ∃ C : ℝ, 0 < C ∧ ∃ c : ℝ, 0 < c ∧
      ∀ A : Finset ℤ, 2 ≤ A.card → ((A + A).card : ℝ) ≤ K * (A.card : ℝ) →
        c * (A.card : ℝ) ^ 2 / (Real.log (A.card : ℝ)) ^ C ≤ ((A * A).card : ℝ) := by
…

    Mentions: c

  2. theorem Erdos818.erdos_818.variants.solymosi This was proved by Solymosi [So09d], in the strong form $$\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{\log \lvert A\rvert}.$$ sorry line 56 source

    Source documentation

    This was proved by Solymosi [So09d], in the strong form $$\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{\log \lvert A\rvert}.$$

    depends on sorry what this means

    Statement uses: Real, LT.lt, OfNat.ofNat, Exists, And, Finset, Int, LE.le, Nat, Finset.card, HAdd.hAdd, HMul.hMul, HDiv.hDiv, HPow.hPow, Real.log

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

    46 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_818.variants.solymosi :
    ∀ K : ℝ, 0 < K → ∃ c : ℝ, 0 < c ∧
      ∀ A : Finset ℤ, 2 ≤ A.card → ((A + A).card : ℝ) ≤ K * (A.card : ℝ) →
        c * (A.card : ℝ) ^ 2 / Real.log (A.card : ℝ) ≤ ((A * A).card : ℝ) := by
…

    Mentions: c