On Two-Dimensional Sums and Differences

The present paper deals with a generalization of a well-known theorem for a set $A \subseteq G$, where $G$ is an arbitrary Abelian group. According to this classical result, it follows from $|A+A|< (3/2) |A|$ or $|A-A| < (3/2) |A|$ that $A \subseteq H$, where $H$ is a coset with respect to some subgroup of $G$ and $|H| \le (3/2) |A|$.
Consider the sets $A^2 \pm \Delta(A) \subseteq G^2$ (two-dimensional sum and difference). Here $A^2 = A \times A$ is the set of pairs of elements from $A$ and $\Delta(A)$ is the diagonal set $\Delta(A) = \{(a, a) \in G \times G \mid a \in A\}$. The main result involves the given sets and is as follows. If $|A^2 \pm \Delta(A)| < 7/4|A|^2$, then $A \subseteq H + x$ for some $x \in G$ and subgroup $H \subseteq G$, where $|H| < 3/2 |A|$.