A Borsuk–Ulam theorem for k(Zp)-actions on products of (modp) homology spheres

Let p be a prime number. It is proven that for the product of free actions of Zp on modp homology spheres Nni, i=1,…,k, where nis are assumed to be odd if p is odd, for any continuous map the set has dimension at least n1+⋯+nk−m(pk−1), provided ni⩾mpi−1(p−1) for all i (1⩽i⩽k).