  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
$\varepsilon<1/3$. Then we describe an explicit (non-periodic) construction for
$\varepsilon =1/3-1/48$. Finally, we use a compactness argument combined with
some ideas from additive combinatorics to show that a finite covering exists
for $\varepsilon =1/5$. The question whether $\varepsilon$ can be arbitrarily
small remains open.
