Subsets of $\mathbb{F}_p^n\times\mathbb{F}_p^n$ without L-shaped configurations

Abstract

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 Szemer\'edi theorem for a two-dimensional four-point configuration in any setting.