Small asymmetric sumsets in elementary abelian 2-groups

Let AA and BB be subsets of an elementary abelian 22-group GG, 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||A+B|<|A|+|B|, then either A+B=GA+B=G, or the complement of A+BA+B in GG is contained in a coset of a subgroup of index at least 88 (whence |A+B|≥78|G|). We indicate conditions for the containment to be strict, and establish a refinement for the case where the sizes of AA and BB differ significantly.