A solution to the Pyjama Problem

TL;DR

This paper proves that finitely many rotations of a vertical stripe around the origin can cover the entire plane for any positive stripe width, using methods related to Furstenberg's topological dynamics theorem.

Contribution

It provides a positive solution to the Pyjama Problem for all positive stripe widths, connecting it to Furstenberg's $ imes 2, imes 3$ Theorem.

Findings

Finitely many rotations suffice to cover the plane.

The problem reduces to a topological dynamics statement.

The proof is analogous to Furstenberg's theorem.

Abstract

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.