A Variant of The Corners Theorem

Abstract

The Corners Theorem states that for any $\alpha >0$ there exists an $N_0$ such that for any abelian group $G$ with $|G|=N\ge N_0$ 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 $d\ne 0$ 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 $d\ne 0$, 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 $|S_d|>({\alpha }^4-\epsilon )N^2$.

17 pages

References in corpus (1)

• Triforce and Corners

Cited by in corpus (2)

• Khintchine-type recurrence for 3-point configurations
• Bounds in a popular multidimensional nonlinear Roth theorem