Green's Open Problem 52

Suppose that A ⊂ 𝔽_2^n is a set with an additive complement of size K. Does 2A contain a coset of codimension O_K(1)?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-green-52,
  title        = {Green's Open Problem 52},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-52}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Suppose that $A \subset F_{2}^{n}$ is a set with an additive complement of size $K$ . Does $2 A$ contain a coset of codimension $O_{K} (1)$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«52». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_52 :
    answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)),
      S.card = K → A + (S : Set (𝔽₂ n)) = Set.univ →
      ∃ (V : AffineSubspace (ZMod 2) (𝔽₂ n)), (V : Set (𝔽₂ n)) ⊆ A + A ∧
        n ≤ Module.finrank (ZMod 2) V.direction + c K

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Green's Open Problems

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.