(n,m)-Fold Covers of Spheres

الأصل · EN

A well known consequence of the Borsuk-Ulam theorem is that if the d-dimensional sphere Sᵈ is covered with less than d+2 open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper bounds on the minimum number of open sets, not containing a pair of antipodal points, needed to cover the d-dimensional sphere n times, with the additional property that the northern hemisphere is covered m > n times. We prove that if the open northern hemisphere is to be covered m times then at least d-1/2 +n+m and at most d+n+m sets are needed. For the case of n=1 and d ≥ 2, this number is equal to d+2 if m ≤ d/2 + 1 and equal to d-1/2 + 2 +m if m > d/2 + 1. If the closed northern hemisphere is to be covered m times then d+2m-1 sets are needed, this number is also sufficient. We also present results on a related problem of independent interest. We prove that if Sᵈ is covered n times with open sets, not containing a pair of antipodal points, then there exists a point that is covered at least d/2 +n times. Furthermore, we show that there are covers in which no point is covered more than n+d times.