On an Argument of Shkredov on Two-Dimensional Corners

Let $F_2^n$ be the finite field of cardinality $2^n$. For all large $n$, any subset $A subset F_2^n imes F_2^n$ of cardinality $$|A| gtrsim 4^n frac{log log n}{log n}$$ must contain three points ${(x,y),(x + d,y),(x,y + d)}$ for $x,y,d in F_2^n$ and $d eq 0$. Our argument is an elaboration of an argument of Shkredov, building upon the finite field analog of Ben Green. The interest in our result is in the exponent on $log n$, which is larger than has been obtained previously.