The Minimum Size of Signed Sumsets

Abstract

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho (G,m,h)=\min \{|hA|:A\subseteq G,|A|=m\}$$ and $${\rho }_{\pm }(G,m,h)=\min \{|h_{\pm }A|:A\subseteq G,|A|=m\},$$ where $hA$ and $h_{\pm }A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $\rho (G,m,h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that ${\rho }_{\pm }(G,m,h)$ equals $\rho (G,m,h)$ when $G$ is cyclic, and establish an upper bound for ${\rho }_{\pm }(G,m,h)$ that we believe gives the exact value for all $G$, $m$, and $h$.