Quantitative pyjama

The “pyjama stripe” with parameter $\epsilon >0$ is the set $E(\epsilon )$ of all complex numbers $z$ such that the distance from $\mathrm{\Re }(z)$ to the nearest integer is at most $\epsilon$. The Pyjama Problem of Iosevich, Kolountzakis, and Matolcsi asks whether, for every choice of $\epsilon >0$, it is possible to cover the entire complex plane with finitely many rotations of $E(\epsilon )$ 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 $\epsilon$) on the number of rotations required, and Green has highlighted the problem of obtaining such quantitative bounds. Our main result is that $\exp \exp \exp ({\epsilon }^{-O(1)})$ rotations of $E(\epsilon )$ 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.