On the Multiplicity of Arrangements of Congruent Zones on the Sphere

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 Naszódi [8]. 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.