Small sumsets in free products of ℤ/2ℤ

Abstract

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 ε A, b ε 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 Gi, except if the factors Gi are all isomorphic to ℤ, in which case μG (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 Gi are all isomorphic to ℤ/2ℤ, and prove that μG (r,s) = r + s - 2 or r + s -1, depending on whether r and s are both even or not. © 2010 Springer Science+Business Media, LLC.