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