A solution to the pyjama problem (Q887952)

For a real number \(x\), let \(\|x\|\) denote the distance to a closest integer. Given \(\varepsilon > 0\), define the set of equidistant planar vertical strips of width \(2\varepsilon\), called the pyjama stripes, as \[ E(\varepsilon) := \left\{ (x,y) \in \mathbb R^2 : \|x\| \leq \varepsilon \right\}. \] The pyjama problem then asks whether the plane \(\mathbb R^2\) can be covered by finitely many rotations of \(E(\varepsilon)\)? In other words, if \(R_{\theta}\) stands for counterclockwise \(2 \times 2\) rotation by the angle of \(\theta\), does there exist a finite collection of angles \(\theta_1,\dots,\theta_n\) such that \[ \mathbb R^2 = R_{\theta_1} E(\varepsilon) \cup \dots \cup R_{\theta_1} E(\varepsilon)? \] The paper under review answers this question in the affirmative for all \(\varepsilon > 0\). The proof proceeds by reducing this problem to a statement resembling Furstenberg's \(\times 2\), \(\times 3\) Theorem from topological dynamics, which is proved by analogy with this result.