Let A be a subset of an n dimensional vector space over the field with two elements, and define A as the set of sums of two elements of A. Naively, the more independent the elements of A, the larger is 2A relative to A; for instance if A is just m independent elements, then 2A will contain 1+m(m-1)/2 elements, whereas the cardinalities are equal if and only if A is a coset of a subspace.

This paper considers the case when the ratio of the cardinalities, say s, is less than 2. It proves and shows optimal the result that, if without loss of generality 0 is in A, the ratio of the cardinality of the subspace generated by A to that of A is s (s<1.75) and < 8s/7 (1.75<=s<2). It does this by showing the existence of a group H containing 2A with H\2A trivial (always the case if n is 1 or 2), or a full P coset where P is a subgroup of H of index at least 8.

This mathematical combinatorial research paper makes explicit what is contained in a number of other papers, and provides a good overview of the area and a solid set of references. The proofs, which constitute the bulk of the article, are well laid out and easy to follow. It would have been nice to see some coding applications discussed, but the omission is probably one of culture.