FormalConjectures/ErdosProblems/245.lean

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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 245

Reference: erdosproblems.com/245

Imports (1)
  1. theorem Erdos245.erdos_245 Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$ sorry line 46 source

    Source documentation

    Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$. Is it true that $$ \limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 3? $$

    The answer is yes, proved by Freiman [Fr73].

    [Fr73] Fre\u{\i}man, G. A., Foundations of a structural theory of set addition. (1973), vii+108.

    depends on sorry what this means

    Statement uses: Iff, True, Set, Nat, Set.Infinite, Filter.Tendsto, Real, HDiv.hDiv, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Nat.floor, Filter.atTop, nhds, LE.le, EReal, Filter.limsup, HAdd.hAdd, Set.add

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

    72 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_245 :
    answer(True) ↔ ∀ (A : Set ℕ), A.Infinite →
      atTop.Tendsto (fun N ↦ (A ∩ Icc 1 ⌊N⌋₊ |>.ncard : ℝ) / N) (𝓝 0) →
      3 ≤ atTop.limsup
        fun N : ℝ ↦ ((A + A) ∩ Icc 1 ⌊N⌋₊ |>.ncard : EReal)
          / (A ∩ Icc 1 ⌊N⌋₊).ncard := by
…
  2. theorem Erdos245.erdos_245.variants.two Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$ sorry line 62 source

    Source documentation

    Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$. Then $$ \limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 2. $$

    depends on sorry what this means

    Statement uses: Set, Nat, Set.Infinite, Filter.Tendsto, Real, HDiv.hDiv, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Nat.floor, Filter.atTop, nhds, LE.le, EReal, Filter.limsup, HAdd.hAdd, Set.add

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

    72 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_245.variants.two (A : Set ℕ) (h_inf : A.Infinite)
    (hf : atTop.Tendsto (fun N ↦ (A ∩ Icc 1 ⌊N⌋₊ |>.ncard : ℝ) / N) (𝓝 0)) :
    2 ≤ atTop.limsup
      fun N : ℝ ↦ ((A + A) ∩ Icc 1 ⌊N⌋₊ |>.ncard : EReal)
        / (A ∩ Icc 1 ⌊N⌋₊).ncard := by
…