Small Sumsets in Free Products of $$\mathbb{Z}/2\mathbb{Z}$$

Summary Let G be a group. For positive integers r, s ≤ | G |, let μ G (r, s) denote the smallest possible size of a sumset (or product set) AB = { ab∣a ∈ A, b ∈ B} for any subsets A, B ⊂ G subject to | A | = r, | B | = s. The behavior of μ G (r, s) is unknown for the free product G of groups G i , except if the factors G i are all isomorphic to $$\mathbb{Z}$$ , in which case $${\mu }_{G}(r,s) = r + s - 1$$ by a theorem of Kemperman for torsion-free groups (1956). In this paper, we settle the case of a free product G whose factors G i are all isomorphic to $$\mathbb{Z}/2\mathbb{Z}$$ , and prove that $${\mu }_{G}(r,s) = r + s - 2$$ or $$r + s - 1$$ , depending on whether r and s are both even or not.