$(n,m)$-fold covers of spheres

A well-known consequence of the Borsuk–Ulam theorem is that if the $d$-dimensional sphere $S^d$ 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 $\lceil(d-1)/2\rceil+n+m$ and at most $d+n+m$ sets are needed. For the case of $n=1$ and $d\ge2$, this number is equal to $d+2$ if $m\le\lfloor d/2\rfloor+1$ and equal to $\lfloor(d-1)/2\rfloor+2+m$ if $m>\lfloor d/2\rfloor+1$. 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 $S^d$ is covered $n$ times with open sets not containing a pair of antipodal points, then there exists a point that is covered at least $\lceil d/2\rceil+n$ times. Furthermore, we show that there are covers in which no point is covered more than $n+d$ times.