On the multiplicity of arrangements of congruent zones on the sphere

TL;DR

This paper investigates arrangements of congruent zones on high-dimensional spheres, demonstrating that for large n, such zones can be arranged to limit overlaps and coverages, extending previous 3D results.

Contribution

It proves new bounds on the overlap and coverage of congruent zones on spheres in higher dimensions, generalizing prior 3D findings.

Findings

Existence of arrangements with bounded overlaps for large n

Coverage of spheres with zones where each point belongs to at most A_d ln n zones

Extension of 3D results to higher dimensions

Abstract

Consider an arrangement of $n$ congruent zones on the $d$-dimensional unit sphere $S^{d-1}$, where a zone is the intersection of an origin symmetric Euclidean plank with $S^{d-1}$. We prove that, for sufficiently large $n$, it is possible to arrange $n$ congruent zones of suitable width on $S^{d-1}$ such that no point belongs to more than a constant number of zones, where the constant depends only on the dimension and the width of the zones. Furthermore, we also show that it is possible to cover $S^{d-1}$ by $n$ congruent zones such that each point of $S^{d-1}$ belongs to at most $A_d\ln n$ zones, where the $A_d$ is a constant that depends only on $d$. This extends the corresponding $3$-dimensional result of Frankl, Nagy and Nasz\'odi (2016). Moreover, we also examine coverings of $S^{d-1}$ with congruent zones under the condition that each point of the sphere belongs to the interior of at most $d-1$ zones.