A solution to the Pyjama Problem

The "pyjama stripe" is the subset of $\mathbb{R}^2$ consisting of a vertical strip of width $2 \varepsilon$ around every integer $x$-coordinate. The "pyjama problem" asks whether finitely many rotations of the pyjama stripe around the origin can cover the plane. The purpose of this paper is to answer this question in the affirmative, for all positive $\varepsilon$. The problem is reduced to a statement closely related to Furstenberg's $\times 2,\, \times 3$ Theorem from topological dynamics, and is proved by analogy with that result.