论文标题
$ \ mathbb {f} _p^n \ times \ mathbb {f} _p^n $的子集
Subsets of $\mathbb{F}_p^n\times\mathbb{F}_p^n$ without L-shaped configurations
论文作者
论文摘要
修复Prime $ P \ GEQ 11 $。我们表明存在一个正整数$ m $，使得$ \ mathbb {f} _p^n \ times \ times \ mathbb {f} _p^n $的任何子集都不包含$ $ $ $ $ $ $（x，y）的非散布配置1/\ log_ {m} {n} $，其中$ \ log_ {m} $表示$ m $ - 折叠的迭代对数。这给出了多维Szemerédi定理中的第一个合理绑定，用于在任何情况下进行二维四点配置。
Fix a prime $p\geq 11$. We show that there exists a positive integer $m$ such that any subset of $\mathbb{F}_p^n\times\mathbb{F}_p^n$ containing no nontrivial configurations of the form $(x,y),(x,y+z),(x,y+2z),(x+z,y)$ must have density $\ll 1/\log_{m}{n}$, where $\log_{m}$ denotes the $m$-fold iterated logarithm. This gives the first reasonable bound in the multidimensional Szemerédi theorem for a two-dimensional four-point configuration in any setting.
