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 ⊆ G, |A| = m} and ρ±(G,m, h) = min{|h±A| : A ⊆ G, |A| = m}, where hA and h±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±(G,m, h) equals ρ(G,m, h) when G is cyclic, and establish an upper bound for ρ±(G,m, h) that we believe gives the exact value for all G, m, and h.