A note on the pyjama problem

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 εε. We first prove that there exist no periodic coverings for ε<13. Then we describe an explicit (non-periodic) construction for ε=13−148. Finally, we use a compactness argument combined with some ideas from additive combinatorics to show that finite coverings exist for all ε>15. The question whether εε can be arbitrarily small remains open.