On the subgroup generated by a small doubling binary set

Let A be a subset of (Z/2Z)n, such that |2A|<2|A|. In this paper, we prove that there exist a subgroup H of (Z/2Z)n and a subgroup P of H with |P|≤|H|/8 such that H contains 2A, and H⧹2A 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.