A variant of the Corners theorem

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