On an Argument of Shkredov on Two-Dimensional Corners

Let ${\mathbb{F}}_2^n$ be the finite field of cardinality $2^n$. For all large $n$, any subset $A\subset {\mathbb{F}}_2^n\times {\mathbb{F}}_2^n$ of cardinality \begin{equation*} \abs{A} \gtrsim 4^n \log\log n (\log n) ^{-1} \end{equation*} must contain three points $\{(x,y),(x+d,y),(x,y+d)\}$ for $x,y,d\in {\mathbb{F}}_2^n$ and $d\mathrm{\ne }0$. Our argument is an elaboration of an argument of Shkredov \cite {math.NT/0405406}, building upon the finite field analog of Ben Green \cite {math.NT/0409420}. The interest in our result is in the exponent on $\log n$, which is larger than has been obtained previously.