Geodesic
On Two-Dimensional Sums in Abelian Groups
Matematičeskie zametki, Tome 103 (2018) no. 2, pp. 273-294

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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, $|H| (3/2)|A|$.
Keywords: Abelian group, linear operator, sums of sets, additive combinatorics.
Mots-clés : convolution 
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     author = {A. A. Uvakin},
     title = {On {Two-Dimensional} {Sums} in {Abelian} {Groups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {273--294},
     publisher = {mathdoc},
     volume = {103},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_103_2_a9/}
}
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A. A. Uvakin. On Two-Dimensional Sums in Abelian Groups. Matematičeskie zametki, Tome 103 (2018) no. 2, pp. 273-294. http://geodesic.mathdoc.fr/item/MZM_2018_103_2_a9/