A note on the pyjama problem

• Main Authors: Malikiosis, R.D., Matolcsi, M., Ruzsa, I.Z.
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2013
• Subjects: DRNTU::Science::Physics
• Online Access: https://hdl.handle.net/10356/106094; http://hdl.handle.net/10220/17933; http://dx.doi.org/10.1016/j.ejc.2013.03.001
• Institution: Nanyang Technological University

This note concerns the so-called pyjama problem, whether it is possible to cover the plane by finitely many rotations of vertical strips of half-width ε. We first prove that there exist no periodic coverings for ε<1/3. Then we describe an explicit (non-periodic) construction for ε=1/3 - 1/48. Finally, we use a compactness argument combined with some ideas from additive combinatorics to show that finite coverings exist for all ε>1/5. The question whether ε can be arbitrarily small remains open.