  Let $A$ and $B$ be additive sets of $\mathbb{Z}_{2k}$, where $A$ has
cardinality $k$ and $B=v.\complement A$ with $v\in\mathbb{Z}_{2k}^{\times}$. In
this note some bounds for the cardinality of $A+B$ are obtained, using four
different approaches. We also prove that in a special case the bound is not
sharp and we can recover the whole group as a sumset.
