Ben Green's Open Problem 61

Suppose that A + A contains the first n squares. Is |A| ≥ n^1 - o(1)? It is known that necessarily |A| ≥ n^2/3 - o(1), whilst in the other direction there do exist such A with |A| ≪_C n / log^C n for any 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-61,
  title        = {Ben Green's Open Problem 61},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-61}},
  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

Suppose that $A + A$ contains the first $n$ squares. Is $∣ A ∣ \geq n^{1 - o (1)}$ ?

It is known that necessarily $∣ A ∣ \geq n^{2/3 - o (1)}$ , whilst in the other direction there do exist such $A$ with $∣ A ∣ ≪_{C} n / lo g^{C} n$ for any $C$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«61». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_61 :
    answer(sorry) ↔
      ∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧
        ∀ n : ℕ, n ≥ 1 → ∀ (A : Finset ℕ),
          (Finset.Icc 1 n).image (· ^ 2) ⊆ A + A →
            (n : ℝ) ^ (1 - f n) ≤ A.card

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 61](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 61)

This problem was originally considered by Erdős and Newman.

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.