A note on a sumset in ${\mathbb{Z}}_{2k}$

Abstract

Let $A$ and $B$ be additive sets of ${\mathbb{Z}}_{2k}$, where $A$ has cardinality $k$ and $B=v\cdot CA$ 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.