(n,m)-Fold Covers of Spheres

A well known consequence of the Borsuk-Ulam theorem is that if the d-dimensional sphere S 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 S 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.