A note on the pyjama problem

TL;DR

This paper investigates the pyjama problem, exploring conditions under which the plane can be covered by finitely many rotated strips of certain widths, providing new constructions and non-existence results for specific widths.

Contribution

It proves the non-existence of periodic coverings for widths less than 1/3, constructs a non-periodic covering for a specific width, and shows a finite covering exists for another width using advanced mathematical techniques.

Findings

No periodic coverings for ε<1/3

Explicit non-periodic covering for ε=1/3-1/48

Finite covering exists for ε=1/5

Abstract

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.