Source labels openChecked July 26, 2026

Ben Green's Open Problem 29

Suppose that $A$ is a $K$ -approximate group (not necessarily abelian). Is there $S \subset A$ , $|S| \gg K^{-O(1)} |A|$ , with $S^8 \subset A^4$ ?

Group theory AMS 20

Mathematical statement

Suppose that $A$ is a $K$ -approximate group (not necessarily abelian). Is there $S \subset A$ , $|S| \gg K^{-O(1)} |A|$ , with $S^8 \subset A^4$ ?

Statement source: Green's Open Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

green_29

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem green_29 :    answer(sorry) ↔      ∃ C c : ℝ, 0 < C ∧ 0 < c ∧        ∀ {G : Type*} [Group G] [DecidableEq G] (K : ℝ) (A : Finset G),          1 ≤ K → IsApproximateSubgroup K (A : Set G) →            ∃ S ⊆ A, C * K ^ (-c) * (A.card : ℝ) ≤ (S.card : ℝ) ∧            S ^ 8 ⊆ A ^ 4 := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Ben Green's Open Problem 29background