  Let $V$ and $W$ be orthogonal representations of $G$ with $V^G= W^G=\{0\}$.
Let $S(V )$ be the sphere of $V$ and $f : S(V ) \to W$ be a $G$-equivariant
mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\{0\}$ in
terms of $ \dim V$ and $\dim W$, if $G$ is the torus $\mathbb T^k$, or the
$p$-torus $\mathbb Z_p^k$. This extends the classical Bourgin-Yang theorem onto
this class of groups. Finally, we show that for any $p$-toral group $G$ and a
$G$-map $f:S(V) \to W$, with $\dim V=\infty$ and $\dim W<\infty$, we have $\dim
Z_f= \infty$.
