Quantitative pyjama

The "pyjama stripe" with parameter $\varepsilon>0$ is the set $E(\varepsilon)$ of all complex numbers $z$ such that the distance from $\Re(z)$ to the nearest integer is at most $\varepsilon$. The Pyjama Problem of Iosevich, Kolountzakis, and Matolcsi asks whether, for every choice of $\varepsilon>0$, it is possible to cover the entire complex plane with finitely many rotations of $E(\varepsilon)$ around the origin. Manners obtained an affirmative answer to this question by studying a $\times 2, \times 3$-type problem over a suitable solenoid. Manners's argument provided no quantitative bounds (in terms of $\varepsilon$) on the number of rotations required, and Green has highlighted the problem of obtaining such quantitative bounds. Our main result is that $\exp\exp\exp(\varepsilon^{-O(1)})$ rotations of $E(\varepsilon)$ suffice to cover the complex plane. Our analysis makes use of the entropic tools developed by Bourgain, Lindenstrauss, Michel, and Venkatesh for quantitative $\times 2, \times 3$-type results.