Geodesic
On a problem of Gowers
Izvestiya. Mathematics , Tome 70 (2006) no. 2, pp. 385-425

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We prove that every set $A\subseteq\{1,\dots,N\}^2$ of cardinality at least $\delta N^2$ contains a triple of the form $\{(k,m),(k+d,m),(k,m+d)\}$, where $d>0$, $\delta>0$ is any real number, $N$ is a positive integer, $N\geqslant \exp\exp\exp\{\delta^{-c}\}$, and $c>0$ is an effective constant. 
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     author = {I. D. Shkredov},
     title = {On a problem of {Gowers}},
     journal = {Izvestiya. Mathematics },
     pages = {385--425},
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     number = {2},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2006_70_2_a6/}
}
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I. D. Shkredov. On a problem of Gowers. Izvestiya. Mathematics , Tome 70 (2006) no. 2, pp. 385-425. http://geodesic.mathdoc.fr/item/IM2_2006_70_2_a6/