On the subgroup generated by a small doubling binary set

Let A be a subset of (double-struck Z/2 double-struck Z)ⁿ, such that 2A < 2 A . In this paper, we prove that there exist a subgroup H of (double-struck Z/2 double-struck Z)ⁿ and a subgroup P of H with P ≤ H /8 such that H contains 2A, and H2A is either empty or a full P-coset. We use this result to obtain an upper bound for the cardinality of the subgroup 〈A〉 generated by A in terms of A . More precisely we show that if 0∈A and 2A =τ A then 〈A〉 / A is equal to τ if 1≤τ<7/4, and is less than 8τ/7 if 7/4≤τ<2. This result is optimal.