Quantitative pyjama

TL;DR

This work resolves the Pyjama Problem by providing an explicit, though large, quantitative bound: the complex plane can be covered by $\exp\exp\exp(\varepsilon^{-C})$ rotations of the pyjama stripe $E(\varepsilon)$ for small $\varepsilon$. The authors develop a dynamical framework on a compact dual space $X$ to mimic Furstenberg’s $\times 2,\times 3$ theory, and they adapt entropic techniques from Bourgain–Lindenstrauss–Michel–Venkatesh to a $X$-setting, achieving a quantitative rationality dichotomy: either an orbit is $\varepsilon$-dense or the point lies near a torsion ball of controlled order. The backbone combines major/minor arc analyses, Baker-type Diophantine bounds, and entropy arguments, then translates the density information back to the plane via an irrational-rotation trick, culminating in a finite, explicit collection of rotations that covers $\mathbb{C}$. The results illustrate the power of entropic dynamics in geometric covering problems and connect to broader themes such as the Lonely Runner problem through dilation-rotation variants.

Abstract

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.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2: Quantitative rationality lemma
• Theorem 2.1
• Theorem 2.2: Manners's rationality lemma
• Theorem 2.3
• Theorem 2.4
• proof : Proof of Theorem \ref{['thm:quantitative-furstenberg']}
• Lemma 3.1
• proof
• Lemma 3.2