Let $A$ be a finite subset of an abelian group $G$. For every
element $b_i$ of the sumset $2A=\{b_0, b_1, ...,b_{|2A|-1}\}$ we
denote by $D_i=\{a-a': a, a' \in A; a+a'=b_i\}$ and
$r_i=|\{(a,a'): a+a'=b_i; a, a' \in A \}|$.
After an eventual reordering of $2A$, we may assume that $r_0\geq
r_1 \geq ...\geq r_{|2A|-1}.$ For every $1 \le s \le |2A|$ we
define $R_s(A)=|D_0\cup D_1\cup...\cup D_{s-1}|$ and $R_s(k)=\max
\{R_s(A): A\subseteq G, |A| =k\}.$ Bourgain and Katz and Tao
obtained an estimate of $R_s(k)$ assuming $s$ being of order $k$.
In this paper we describe the {\it structure} of $A$ assuming that\break
$G=\mathbb{Z}^2, s=3$ and $R_3(A)$ is close to its maximal value,
i.e. $R_3(A) = 3k-\theta \sqrt{k}$, with $\theta \le 1.8$.