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Fix a prime $p\geq 11$. We show that there exists a positive integer $m$ suchthat any subset of $\mathbb{F}_p^n\times\mathbb{F}_p^n$ containing nonontrivial configurations of the form $(x,y),(x,y+z),(x,y+2z),(x+z,y)$ musthave density $\ll 1/\log_{m}{n}$, where $\log_{m}$ denotes the $m$-folditerated logarithm. This gives the first reasonable bound in themultidimensional Szemer\'edi theorem for a two-dimensional four-pointconfiguration in any setting.

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