Quantitative pyjama

TL;DR

This paper establishes a quantitative bound on the number of rotations of the pyjama stripe needed to cover the complex plane, advancing the understanding of the Pyjama Problem with explicit bounds.

Contribution

It provides the first explicit exponential bound on the number of rotations required to cover the plane in the Pyjama Problem, using entropic methods.

Findings

Bound of triple exponential in epsilon for covering the plane.

Application of entropic tools from additive combinatorics.

Extension of previous qualitative results to quantitative bounds.

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.