  Let G be an arbitrary Abelian group and let A be a finite subset of G. A has
small additive doubling if |A+A| < K|A| for some K>0. These sets were studied
in papers of G.A. Freiman, Y. Bilu, I. Ruzsa, M.C.--Chang, B. Green and T.Tao.
In the article we prove that if we have some minor restrictions on K then for
any set with small doubling there exists a set Lambda, |Lambda| << K log |A|
such that |A\cap Lambda| >> |A| / K^{1/2 + c}, where c > 0. In contrast to the
previous results our theorem is nontrivial for large K. For example one can
take K equals |A|^\eta, where \eta>0. We use an elementary method in our proof.
