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Level A · Machine-checkable Hard Geometry P-green-41

Ben Green's Open Problem 41

How many rotated (about the origin) copies of the 'pyjama set' \(x, y) ∈ ℝ^2 : dist(x, ℤ) ≤ ε\ are needed to cover ℝ^2? That is, determine the minimal number of rotations as a function of ε > 0.

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.

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Cite 
@misc{cairn-green-41,
  title        = {Ben Green's Open Problem 41},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-41}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
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Verified
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Disputed
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Refuted
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On the literature board
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Current state

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

The problem

The question

How many rotated (about the origin) copies of the 'pyjama set' are needed to cover ?

That is, determine the minimal number of rotations as a function of .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«41». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_41 : ∀ ε > 0, minCopies ε = (answer(sorry) : ℝ → ℕ) ε

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.

Variants

  • green_41.variants.double_exponential_bound — Can the triple-exponential bound of [KrLe25] be improved by one exponential?
  • green_41.variants.polynomial_bound — Is ε^-C rotations enough?

References

  • [Gr24] Ben Green's Open Problem 41
  • [Ma15] Manners, Freddie. "A solution to the pyjama problem." Inventiones mathematicae 202.1 (2015): 239-270.
  • [KrLe25] Kravitz, Noah, and James Leng. "Quantitative pyjama." arXiv preprint arXiv:2510.17744 (2025).

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.