A Variant of The Corners Theorem

The Corners Theorem states that for any $\alpha > 0$ there exists an $N_0$ such that for any abelian group $G$ with $|G| = N \geq N_0$ and any subset $A \subset G \times G$ with $|A| \ge \alpha N^2$ we can find a corner in $A$ , i.e. there exist $x, y, d \in G$ with $d \neq 0$ such that $(x, y), (x+d, y), (x, y+d) \in A$. Here, we consider a stronger version: given such a group $G$ and subset $A$, for each $d \in G$ we define $S_d = \{(x, y) \in G \times G : (x, y), (x+d, y), (x, y+d) \in A \}$ . So $|S_d|$ is the number of corners of size $d$. Is it true that, provided $N$ is sufficiently large, there must exist some $d \in G \setminus \{0\}$ such that $|S_d|> (\alpha^3 - \epsilon ) N^2$ ? We answer this question in the negative. We do this by relating the problem to a much simpler-looking problem about random variables. Then, using this link, we show that there are sets $A$ with $|S_d| < C\alpha^{3.13} N^2$ for all $d \neq 0$, where $C$ is an absolute constant. We also show that in the special case where $G = \mathbb{F}_2^n$, one can always find a $d$ with $|S_d|> (\alpha^4 - \epsilon ) N^2$.