Geodesic
On a~two-dimensional analogue of Szemer\'edi's theorem in Abelian groups
Izvestiya. Mathematics , Tome 73 (2009) no. 5, pp. 1033-1075

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Let $G$ be a finite Abelian group and $A\subseteq G\times G$ a set of cardinality at least $|G|^2/(\log\log|G|)^c$, where $c>0$ is an absolute constant. We prove that $A$ contains a triple $\{(k,m),(k+d,m),(k,m+d)\}$ with $d\neq0$. This is a two-dimensional generalization of Szemerédi's theorem on arithmetic progressions.
Keywords: two-dimensional generalizations of Szemerédi's theorem, problems on arithmetic progressions, Roth's theorem, Bohr sets. 
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     author = {I. D. Shkredov},
     title = {On a~two-dimensional analogue of {Szemer\'edi's} theorem in {Abelian} groups},
     journal = {Izvestiya. Mathematics },
     pages = {1033--1075},
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     number = {5},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2009_73_5_a6/}
}
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I. D. Shkredov. On a~two-dimensional analogue of Szemer\'edi's theorem in Abelian groups. Izvestiya. Mathematics , Tome 73 (2009) no. 5, pp. 1033-1075. http://geodesic.mathdoc.fr/item/IM2_2009_73_5_a6/