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 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.