Pyjama problem

Let ${\displaystyle E(\varepsilon ):=\{z\in \mathbb {C}$ :\mathrm {Re} (z)\in (-\varepsilon ,\varepsilon ){\pmod {1}}\}} be the pyjama stripe of width ${\displaystyle 2\varepsilon }$. Noah Kravitz and James Leng proved that ${\displaystyle \exp \exp \exp(\varepsilon ^{-O(1)})}$ rotations of ${\displaystyle E(\varepsilon )}$ about the origin are sufficient to cover ${\displaystyle \mathbb {C} }$, hence obtaining an explicit upper bound for the pyjama problem.^[4] It remains an open problem to obtain lower bounds for the pyjama problem beyond the trivial volume preserving bound of ${\displaystyle \varepsilon ^{-1}/2}$.^[4][5]

• Tarski's plank problem, on covering bounded sets by stripes