On sets with small doubling property

Suppose that G is an arbitrary Abelian group and A is any finite subset G. A set A is called a set with small sumset if, for some number K, we have |A + A| ≤ K|A|. The structural properties of such sets were studied in the papers of Freiman, Bilu, Ruzsa, Chang, Green, and Tao. In the present paper, we prove that, under certain constraints on K, for any set with small sumset, there exists a set Λ, Λ ≪ ɛK log |A|, such that |A ν Λ| ≫ |A|/K1/2+ɛ, where ɛ > 0. In contrast to the results of the previous authors, our theorem is nontrivial even for a sufficiently large K. For example, for K we can take |A|η, where η > 0. The method of proof used by us is quite elementary.