The Corners theorem states that for any α > 0 there exists an N0 such that for any abelian group G with |G| = N ≥ N0 and any subset A ⊂ G×G with |A| ≥ αN2 we can find a corner in A, i.e. there exist x, y, d ∈ G with d ≠ 0 such that (x,y),(x+d,y),(x,y+d) ∈ A.Here, we consider a stronger version, in which we try to find many corners of the same size. Given such a group G and subset A, for each d ∈ G we define Sd={(x,y) ∈ G × G: (x,y),(x+d,y),(x,y+d) ∈ A}. So |Sd| is the number of corners of size d. Is it true that, provided N is sufficiently large, there must exist some d ∈G\{0} such that |Sd|>(α3-ϵ)N2?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 |Sd|>Cα3.13N2 for all d ≠ 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 |Sd|>(α4-ϵ)N2.

中文翻译：

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Corners 定理的一个变体

Corners 定理指出，对于任何α> 0 存在一个ñ0使得对于任何阿贝尔群G与 |G| =ñ≥ñ0和任何子集一种⊂G×G与 |一种| ≥αN2我们可以找到一个角落一种, 即存在X,是的,d∈G和d≠ 0 使得 (X,是的),(X+d,是的),(X,是的+d) ∈一种.在这里，我们考虑一个更强大的版本，我们尝试找到许多相同大小的角。鉴于这样一个群体G和子集一种, 对于每个d∈G我们定义小号d={(X,是的) ∈G×G: (X,是的),(X+d,是的),(X,是的+d) ∈一种}。所以|小号d| 是大小角的数量d. 是不是真的，提供ñ足够大，一定存在一些d∈G\{0} 这样 |小号d|>(α3-ε)ñ2?我们对这个问题的回答是否定的。我们通过将问题与一个看起来更简单的关于随机变量的问题联系起来来做到这一点。然后，使用这个链接，我们显示有集合一种与 |小号d|>α3.13ñ2对所有人d≠0，在哪里C是一个绝对常数。我们还表明，在特殊情况下$G = {\mathbb{F}}_2^n$，总能找到一个d与 |小号d|>(α4-ε)ñ2.