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 $\varepsilon<1/3$. 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. 
R. D. Malikiosis, M. Matolcsi, I. Z. Ruzsa
math.CO math.MG