47.lean`

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

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

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

Green's Open Problem 47

References:

[Gr24] Ben Green's 100 Open Problems
[GH14] Green, Ben, and Adam J. Harper. "Inverse questions for the large sieve." Geometric and Functional Analysis 24.4 (2014): 1167-1203.
[HV09] Helfgott, Harald Andrés, and Akshay Venkatesh. "How small must ill-distributed sets be." Analytic number theory 2 (2009): 224-234.
[Wa12] Walsh, Miguel N. "The inverse sieve problem in high dimensions." (2012): 2001-2022.
[Wa14] Walsh, Miguel N. "The algebraicity of ill-distributed sets." Geometric and Functional Analysis 24.3 (2014): 959-967.

Imports (1)

FormalConjecturesUtil

theorem Green47.green_47 sorry line 52 source
depends on sorry what this means
Statement uses: Iff, False, Set, Nat, Filter.Eventually, Nat.Prime, LE.le, Set.ncard, ZMod, Set.image, HDiv.hDiv, HAdd.hAdd, OfNat.ofNat, Filter.atTop, Or, Asymptotics.IsBigO, Real, Inter.inter, Set.Iic, Real.sqrt and 12 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
64 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green47.green_47.variants.finite_exceptions In Green and Harper's formulation [GH14, Conjecture 1.7], the quadratic containment is allowed to fail on a finite subset of $A$ sorry line 62 source
Source documentation
In Green and Harper's formulation [GH14, Conjecture 1.7], the quadratic containment is allowed to fail on a finite subset of $A$.

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Filter.Eventually, Nat.Prime, LE.le, Set.ncard, ZMod, Set.image, HDiv.hDiv, HAdd.hAdd, OfNat.ofNat, Filter.atTop, Or, Asymptotics.IsBigO, Real, Inter.inter, Set.Iic, Real.sqrt and 15 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
64 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_47.variants.finite_exceptions :
    answer(sorry) ↔ ∀ A : Set ℕ,
      (∀ᶠ p in atTop, Nat.Prime p → Set.ncard (Set.image (fun a : ℕ => (a : ZMod p)) A) ≤ (p + 1) / 2) →
      ((fun X : ℕ => ((A ∩ Set.Iic X).ncard : ℝ)) ≪ (fun X : ℕ => Real.sqrt (X : ℝ) / (Real.log (X : ℝ)) ^ 100))
      ∨ (∃ P : Polynomial ℚ, P.degree = 2 ∧ {a ∈ A | ¬∃ z : ℤ, (a : ℚ) = P.eval (z : ℚ)}.Finite) := by
…
```