A variant of the Corners theorem
A variant of the Corners theorem
复制标题
角点定理的一种变体
DOI：
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发表时间：
2018
影响因子：
0.8
通讯作者：
Matei Mandache
中科院分区：
文献类型：
--
作者：
Matei Mandache
摘要Corners定理指出，对于任何α> 0在A，即存在x，y，d∈G中，d≠0，这样（x，y），（x+d，y），（x，y+d）∈A。 ，在其中，我们试图找到相同大小的角落。子集a，对于每个d∈G，我们定义sd = {（x，y）∈G×g：（x，y），（x+d，y），（x，y+d）∈A} SD |。负面的方法是将问题与随机变量相关的更简单的问题。我们表明，对于所有d≠0，有| sd |>cα3.13n2的集合，其中c是一个绝对常数。 ，人们总是可以找到| sd |>（α4-ϵ）N2的D。
Abstract The Corners theorem states that for any α > 0 there exists an N 0 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.
三角力和角点
DOI：
10.1017/s0305004119000173
发表时间：
2020
影响因子：
0.8
作者：
FOX, JACOB;SAH, ASHWIN;SAWHNEY, MEHTAAB;STONER, DAVID;ZHAO, YUFEI
通讯作者：
ZHAO, YUFEI