Property of continuous maps from $S^3$ to $\mathbb{C}$. Let $p=e^{2\pi i/m}$ for some integer $m&gt;1$. If $f:S^3\rightarrow\mathbb{C}$ is a continuous map, prove that $\sum_{k=1}^mp^{-k}f(p^kz_1,p^kz_2)=0$ for some point $(z_1,z_2)\in S^3$.

In class we have just learned the Borusk-Ulam theorem for maps from $S^2\rightarrow \mathbb{R}^2$. I also have access to the following fact. There exist no continuous maps $g:S^3\rightarrow S^1$ such that $g(pz_1,pz_2)=pg(z_1,z_2)$ for all $(z_1,z_2)\in S^3$.

I'm just not seeing how to deconstruct this problem. Any help is greatly appreciated.
Call your map $g$ (you mean to sum over k, not i). Is it clear that $g(pz,pw) = p g(z,w)$? If $g$ was always nonzero, then consider the map $h(z,w) = g(z,w)/|g(z,w)|$. This is a map to the circle with $h(pz,pw) = ph(z,w)$. Contradiction
Call your map $g$ (you mean to sum over k, not i). Is it clear that $g(pz,pw) = p g(z,w)$? If $g$ was always nonzero, then consider the map $h(z,w) = g(z,w)/|g(z,w)|$. This is a map to the circle with $h(pz,pw) = ph(z,w)$. Contradiction