  We describe a connective $K$-theory Borsuk--Ulam/Bourgin--Yang theorem for
cyclic groups of order a power of a prime $p$. Consider two finite dimensional
complex representations $U$ and $V$ of the cyclic group $Z /p^{k+1}$ of order
$p^{k+1}$, where $k\geq 0$. For $0\leq l\leq k$, we write $V_l$ for the
subspace of $V$ fixed by the cyclic subgroup of order $p^l$, and require that
the fixed subspace, $V_{k+1}$, be zero and that $V_k$ be non-zero. Put $\delta
(V)=\sum_{l=0}^k p^l dim_C (V_l/V_{l+1})-(p^k-1)$. Then the zero-set of any $Z
/p^{k+1}$-map $S(U) \to V$ from the unit sphere in $U$ (for some invariant
inner product) has covering dimension greater than or equal to $2(dim_C U -
\delta (V)-1)$, if $dim_C U> \delta (V)$.
