On an Argument of Shkredov on Two-Dimensional Corners

Abstract

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 $d\neq0$. 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.