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.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement green_41 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `green_41`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	3 open, 2 solved	Formal Conjectures

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

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 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

How many rotated (about the origin) copies of the 'pyjama set' $(x, y) \in R^{2} : dist (x, Z) \leq ε$ are needed to cover $R^{2}$ ?

That is, determine the minimal number of rotations as a function of $ε > 0$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«41».

theorem green_41 : ∀ ε > 0, minCopies ε = green_41.answer 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.