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Consider an arrangement of $n$ congruent zones on the $d$-dimensional unitsphere $S^{d-1}$, where a zone is the intersection of an origin symmetricEuclidean plank with $S^{d-1}$. We prove that, for sufficiently large $n$, itis possible to arrange $n$ congruent zones of suitable width on $S^{d-1}$ suchthat no point belongs to more than a constant number of zones, where theconstant 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 suchthat 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 alsoexamine coverings of $S^{d-1}$ with congruent zones under the condition thateach point of the sphere belongs to the interior of at most $d-1$ zones.

On the multiplicity of arrangements of congruent zones on the sphere