Small asymmetric sumsets in elementary abelian 2-groups

Let A and B be subsets of an elementary abelian 2 -group G , neither of which is contained in a coset of a proper subgroup. Extending onto potentially distinct summands a result of Hennecart and Plagne, we show that if | A + B | < | A | + | B | , then either A + B = G , or the complement of A + B in G is contained in a coset of a subgroup of index at least 8 (whence | A + B | ≥ 7 8 | G | ). We indicate conditions for the containment to be strict, and establish a refinement for the case where the sizes of A and B differ significantly.