Quantitative bounds for the pyjama problem

Let E(epsilon) be the set of all complex numbers whose real part is within distance epsilon of an integer. The pyjama problem asks if, for every positive epsilon, finitely many rotations of E(epsilon) can cover the entire complex plane. This was answered affirmatively by Manners in 2012. In joint work with Noah Kravitz, we prove that a triple-exponential number of rotations (in terms of the inverse of epsilon) are sufficient.