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Level A · Machine-checkable Hard Analysis P-green-94

Ben Green's Open Problem 94

Let A ⊂ R be a set of positive measure. Does A contain an affine copy of 1, 1/2, 1/4, . . . ?

Status (as of 2026-11-11): Settled: Formal Conjectures marks this statement as solved, with a formal proof (commit 2f91d251c07f).

Current state (tracked automatically)

  • erdosproblems.com lists Erdős problem #120 as open (status updated 31 August 2025).
  • Formal Conjectures marks the Lean statement green_94 as solved.
All tracked facts and sources
FactValueSource
Status of Erdős problem #120 on erdosproblems.comopen
2025-08-31
erdosproblems.com
Formal Conjectures: status of `green_94`research solvedFormal Conjectures
Formal Conjectures: research statements in the file0 open, 2 solvedFormal Conjectures

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

Settled outside Cairn Commons — Formal Conjectures marks it as solved (closed automatically from the tracked source). Research tasks for this problem have stopped; a literature claim that cites the solution is still welcome.

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.

Submit a claim Follow (RSS)
Cite 
@misc{cairn-green-94,
  title        = {Ben Green's Open Problem 94},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-94}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

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Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let A ⊂ R be a set of positive measure. Does contain an affine copy of {1, 1/2, 1/4, . . . }?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«94». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_94 :
   answer(sorry) ↔ ∀ A : Set ℝ,
   MeasurableSet A ∧ volume A > 0 →
   ∃ a b : ℝ, a ≠ 0 ∧ ∀ n : ℕ, a * (1 / 2^n) + b ∈ A

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

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.