FormalConjectures/ErdosProblems/1002.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 1002

References:

Imports (1)
Imported by 1 file
  1. theorem Erdos1002.erdos_1002 For any $0<\alpha<1$, let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\})$ sorry line 42 source

    Source documentation

    For any $0<\alpha<1$, let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\})$. Does $f(\alpha,n)$ have an asymptotic distribution function?

    In other words, is there a non-decreasing function $g$ such that $g(-\infty)=0$, $g(\infty)=1$, and $\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c)$?

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Real, And, Monotone, Filter.Tendsto, Filter.atBot, nhds, OfNat.ofNat, Filter.atTop, Nat, ENNReal.toReal, MeasureTheory.Measure, Set, ENNReal, MeasureTheory.MeasureSpace.volume, Set.ofPred, Membership.mem, Set.Ioo 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 erdos_1002 :
    answer(sorry) ↔
      ∃ g : ℝ → ℝ, Monotone g ∧
      Tendsto g atBot (𝓝 0) ∧
      Tendsto g atTop (𝓝 1) ∧
      letI f :=  fun (α : ℝ) (n : ℕ) ↦
…
  2. theorem Erdos1002.erdos_1002.variants.kesten Documented declaration: kesten sorry line 60 source

    Source documentation

    Kesten [Ke60] proved that if $f(\alpha,\beta,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{\beta+\alpha k\})$ then $f(\alpha,\beta,n)$ has asymptotic distribution function $g(c)=\frac{1}{\pi}\int_{-\infty}^{\rho c}\frac{1}{1+t^2}\mathrm{d}t$, where $\rho>0$ is an explicit constant.

    depends on sorry what this means

    Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, HMul.hMul, HDiv.hDiv, Real.pi, MeasureTheory.integral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.Iic, HAdd.hAdd, HPow.hPow, Nat, Filter.Tendsto, ENNReal.toReal, MeasureTheory.Measure, Prod, Set and 13 more

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

    92 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_1002.variants.kesten :
    ∃ ρ > 0,
      let g := fun (c : ℝ) ↦ (1 / π) * ∫ t in Iic (ρ * c), 1 / (1 + t^2)
…

    Mentions: c