Green's Open Problem 51

Suppose that A ⊂ 𝔽_2^n is a set of density α. What is the largest size of coset guaranteed to be contained in 2A? We phrase this by asking for the exact function F(α, n) giving the maximum dimension of a guaranteed coset.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 4 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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.

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Cite

@misc{cairn-green-51,
  title        = {Green's Open Problem 51},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-51}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

green_51. Suppose that $A \subset F_{2}^{n}$ is a set of density $α$ . What is the largest size of coset guaranteed to be contained in $2 A$ ?

We phrase this by asking for the exact function $F (α, n)$ giving the maximum dimension of a guaranteed coset.

green_51.one_half. Suppose that $A \subset F_{2}^{n}$ has density $α > 1/2 - C / n$ . Does $A + A$ contain a subspace of co-dimension $O_{C} (1)$ ? [Sa11, Question 5.1]

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«51» (2 statements).

theorem green_51 : green_51.answer = guaranteedMaxCosetDim

theorem green_51.one_half :
    ∀ (k : ℝ), 0 < k →
      ∃ (c : ℕ), ∀ᶠ (n : ℕ) in atTop,
        ∀ (α : ℝ), α > (1/2 : ℝ) - k / sqrt (n : ℝ) → α ≤ 1 →
          n ≤ guaranteedMaxCosetDim n α + c

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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

[Gr24] Green's Open Problems
[Gr13] B. J. Green, Restriction and Kakeya phenomena, notes from a 2003 course. Available at http://people.maths.ox.ac.uk/greenbj/papers/rkp.pdf
[Sa11] Sanders, Tom. "Green's sumset problem at density one half." Acta Arithmetica 146.1 (2011): 91-101.
[Gr02] Green, Ben. "Arithmetic progressions in sumsets." Geometric & Functional Analysis GAFA 12.3 (2002): 584-597.
[Ruz91] Ruzsa, Imre Z. "Arithmetic progressions in sumsets." Acta Arithmetica 60.2 (1991): 191-202.

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.