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 $\mathrm{\mid }G\mathrm{\mid }=N\ge N_0$ and any subset $A \subset G \times G$with$|A| \ge \alpha N^2$wecanfindacornerin$A$ , i.e. there exist $x,y,d\in G$ with $d\mathrm{\ne }0$ such that $(x, y), (x+d, y), (x, y+d) \in A$\mathrm{.}Here,weconsiderastrongerversion:givensuchagroup$G$andsubset$A$,foreach$d \in G$wedefine$S_d = {(x, y) \in G \times G : (x, y), (x+d, y), (x, y+d) \in A }$\mathrm{.}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 $\mathrm{\mid }{S}_d\mathrm{\mid }>({\alpha }^3-ϵ)N^2$ ? 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 $\mathrm{\mid }{S}_d\mathrm{\mid }<C{\alpha }^{3.13}{N}^2$ 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 $|S_d|> (\alpha^4 - \epsilon ) N^2$.