Pyjama problem

In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are. The problem was posed in 2006 by Alex Iosevich, Mihail Kolountzakis, and Máté Matolcsi.^[2] It was answered in the affirmative by Freddie Manners in 2015, using an analogy with Furstenberg’s ×2, ×3 Theorem.^[3]

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