How many rotated pyjama sets cover the plane
For $\varepsilon > 0$ the pyjama set is $\{(x, y) \in \mathbb{R}^2 : \operatorname{dist}(x, \mathbb{Z}) \le \varepsilon\}$. Manners proved, by an ineffective argument from topological dynamics, that finitely many rotations of it about the origin cover $\mathbb{R}^2$, and Kravitz and Leng showed, by analysing his method, that $\exp\exp\exp(\varepsilon^{-C})$ rotations suffice. Green asks in particular whether $\varepsilon^{-C}$ rotations are enough.
Problem. How many rotated copies of the pyjama set, rotated about the origin, are needed to cover $\mathbb{R}^2$?
Source: A quantitative form of the pyjama problem of A. Iosevich, M. N. Kolountzakis and M. Matolcsi (2006), solved qualitatively by F. Manners. Listed as Problem 41 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- CO — Combinatorics
- Secondary
- DS, MG
- Keywords
- coverings, pyjama problem, quantitative bounds, topological dynamics
References
@misc{iosevich2006pyjama,
author = {Iosevich, Alex and Kolountzakis, Mihail N. and Matolcsi, M{\'a}t{\'e}},
title = {Covering the plane by rotations of a lattice arrangement of disks},
year = {2006},
eprint = {math/0611800},
archivePrefix = {arXiv}
}
@article{manners2015pyjama,
author = {Manners, Freddie},
title = {A solution to the pyjama problem},
journal = {Inventiones Mathematicae},
volume = {202},
number = {1},
pages = {239--270},
year = {2015},
doi = {10.1007/s00222-014-0571-7}
}
@misc{kravitz2025pyjama,
author = {Kravitz, Noah and Leng, James},
title = {Quantitative pyjama},
year = {2025},
eprint = {2510.17744},
archivePrefix = {arXiv}
}
@misc{hu2026lonely,
author = {Hu, Xiyu and Jing, Yifan},
title = {Near optimal gap in the lonely runner conjecture},
year = {2026},
eprint = {2610.23790},
archivePrefix = {arXiv}
}
@misc{kitamura2026pyjama,
author = {Kitamura, Kenta},
title = {A {L}ean proof of a quasi-polynomial bound in {G}reen's {P}roblem 41},
howpublished = {\url{https://github.com/KitaKen1/green-41-quasi-polynomial}},
year = {2026},
note = {Commit 5f8398a; AI-assisted Lean formalisation, not a paper}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 41; version of January 2026, most recent update December 2025},
year = {2026}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01036, "How many rotated pyjama sets cover the plane", clopen.solutions.