On a problem of Gowers

I. D. Shkredov

Izvestiya. Mathematics , Tome 70 (2006) no. 2, pp. 385-425.

Résumé

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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