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Subsets of
$\mathbb {F}_p^n\times \mathbb {F}_p^n$ without L-shaped configurations
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 04 December 2023, pp. 176-236
- Print publication:
- January 2024
-
- Article
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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.
