Green's Open Problem 24

If A is a set of n integers, what is the maximum number of affine translates of the set lbrace 0,1,3 rbrace that A can contain? Conjectured in [Aa19] p.579: (1/3 + o(1)) n^2.

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-24,
  title        = {Green's Open Problem 24},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-24}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The question

green_24. If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set ${0, 1, 3}$ that $A$ can contain?

Conjectured in [Aa19] p.579: $(\frac{1}{3} + o (1)) n^{2}$ .

conjecture. Conjecture p.579 in [Aa19]: $(\frac{1}{3} + o (1)) n^{2}$ .

References:

Green, Ben. "100 open problems." (2024).
[Aa19] Aaronson, James. "Maximising the number of solutions to a linear equation in a set of integers." Bulletin of the London Mathematical Society 51.4 (2019): 577-594.
[HaL28] Hardy, G. H., and J. E. Littlewood. "Notes on the theory of series (VIII): an inequality." Journal of the London Mathematical Society 1.2 (1928): 105-110.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«24» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_24 : ∀ n, max013AffineTranslates n = (answer(sorry) : ℕ → ℕ) n

theorem conjecture : gamma = 1/3

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.

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.