Uniform nonlinear Szemer\'{e}di theorem for corners in finite fields

TL;DR

This paper establishes an asymptotic count for nonlinear corner configurations in subsets of finite fields, extending combinatorial results to more general polynomial-based geometric patterns.

Contribution

It introduces a uniform nonlinear Szemerédi theorem for corners in finite fields, generalizing previous linear cases to rational functions.

Findings

Asymptotic formula for nonlinear corners in finite fields

Extension of Szemerédi theorem to rational functions

New techniques for counting polynomial configurations

Abstract

Let $P(t),Q(t)\in \mathbb{Q}(t)$ be rational functions such that $P(t),Q(t)$ and the constant function $1$ are linearly independent over $\mathbb{Q}$, we prove an asymptotic formula for the number of the corner configurations $(x_1,x_2),(x_1+P(y),x_2),(x_1,x_2+Q(y))$ in the subsets of $\mathbb{F}_p^2$.