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

Abstract

Fix a prime $p\ge 11$. 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.

arXiv comment61 pages; v2: typos fixed and small changes made to the exposition; v3: referee suggestions incorporated