On Two-Dimensional Sums in Abelian Groups

It is proved that if, for a subset $A$ of a finite Abelian group $G$, under the action of a linear operator $L\colon G^3 \to G^2$, the image $L(A,A,A)$ has cardinality less than $(7/4)|A|^2$, then there exists a subgroup $H \subseteq G$ and an element $x \in G$ for which $A \subseteq H+x$; further, $|H| < (3/2)|A|$.