On Small Sumsets in (ℤ/2ℤ)ⁿ

It is proved that any subset ${\user1{A}}$ of (ℤ/2ℤ)ⁿ, having k elements, such that ${\left| {{\user1{A}} + {\user1{A}}} \right|} = c{\left| {\user1{A}} \right|}$ (with c<4), is contained in a subgroup of order at most u ^−1k where u=u(c)>0 is an explicit function of c which does not depend on k nor on n. This improves by a radically different method the corresponding bounds deduced from a more general result of I. Z. Ruzsa.