The Minimum Size of Signed Sumsets

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 $h$.