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 \[|A| \gtrsim \frac{4^n \log \log n}{\log n}, \] must contain three points \( \{(x, y), (x + d, y), (x, y + d)\} \) for \( x, y, d \in \mathbb{F}_2^n \) and \( d \neq 0 \). Our argument is an elaboration of an argument of Shkredov [14], building upon the finite field analog of Ben Green [10]. The interest in our result is in the exponent on \( \log n \), which is larger than has been obtained previously.