A note on the pyjama problem

Abstract

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 $\epsilon$. We first prove that there exist no periodic coverings for $\epsilon <1/3$. Then we describe an explicit (non-periodic) construction for $\epsilon =1/3-1/48$. Finally, we use a compactness argument combined with some ideas from additive combinatorics to show that a finite covering exists for $\epsilon =1/5$. The question whether $\epsilon$ can be arbitrarily small remains open.

9 pages

References in corpus (1)

• Covering the plane by rotations of a lattice arrangement of disks

Cited by in corpus (1)

• Rotations by roots of unity and Diophantine approximation