Geodesic
On Two-Dimensional Sums and Differences
Matematičeskie zametki, Tome 98 (2015) no. 4, pp. 570-589.

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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 $|H| 3/2 |A|$.
Keywords: two-dimensional sum and difference, Abelian group, coset with respect to a subgroup, symmetry set, Kneser's theorem. 
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A. A. Uvakin. On Two-Dimensional Sums and Differences. Matematičeskie zametki, Tome 98 (2015) no. 4, pp. 570-589. http://geodesic.mathdoc.fr/item/MZM_2015_98_4_a7/

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