Ben Green's Open Problem 60

Is there an absolute constant c > 0 such that, whenever A ⊆ ℕ is a set of squares with |A| ≥ 2, the sumset A + A satisfies |A + A| ≥ |A|^1 + c?

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.

Start working on it Submit a claim Follow

Cite

@misc{cairn-green-60,
  title        = {Ben Green's Open Problem 60},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-60}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Is there an absolute constant $c > 0$ such that, whenever $A \subseteq N$ is a set of squares with $∣ A ∣ \geq 2$ , the sumset $A + A$ satisfies $∣ A + A ∣ \geq ∣ A ∣^{1 + c}$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«60». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_60 :
    answer(sorry) ↔ ∃ c > (0 : ℝ),
      ∀ (A : Finset ℕ),
        (∀ a ∈ A, IsSquare a) →
        2 ≤ A.card →
          ((A + A).card : ℝ) ≥ (A.card : ℝ) ^ (1 + c)

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

[Ben Green's Open Problem 60](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 60)

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.