The Minimum Size of Signed Sumsets

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

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