A Borsuk-Ulam theorem for $({\mathbb{Z}}_p{)}^k$-actions on products of (mod $p$) homology spheres

It is proved that for a product action of $({\mathbb{Z}}_p{)}^k$ on a product of (mod p) homology spheres $N^{n_1}\times \mathrm{.}\mathrm{.}\mathrm{.}\times N^{n_k}$, where all $n_i$’s are assumed to be odd if $p$ is odd, and any continuous map $f\colon N^{n_1}\times…\times N^{n_k}\to \mathbb R^m$theset$A(f)={x\in N^{n_1}\times…\times N^{n_k}| f(x)=f(gx) \forall g\in(\mathbb Z_p)^k}$ has dimension at least $n_1+\mathrm{.}\mathrm{.}\mathrm{.}+n_k-m(p^k-1)$, provided $n_i\ge m{p}^{i-1}(p-1)$ for all $i(1\le i\le k)$. Moreover, if $n_i\ge m{p}^{k-1}(p-1)$ for all $i(1\le i\le k)$thenthefreeaction$\mu$ can be assumed arbitrary.