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 $|A+A|\le K|A|$. 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 $\Lambda$, $\Lambda\ll_{\varepsilon}K\log|A|$, such that $|A\cap \Lambda|\gg |A|/K^{1/2+\varepsilon}$, where $\varepsilon>0$. 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 $|A|^\eta$, where $\eta>0$. The method of proof used by us is quite elementary.