Uniform nonlinear Szemerédi theorem for corners in finite fields

TL;DR

This work proves a quantitative, uniform two-dimensional Szemerédi-type theorem for corners in finite fields, counting configurations $\bigl(x_1,x_2\bigr),(x_1+P(y),x_2)$ and $\bigl(x_1,x_2+Q(y)\bigr)$ with $P,Q \in \mathbb{Q}(t)$ under the linear independence condition with the constant function $1$. The authors introduce an algebraic-geometry PET induction framework, leverage a Roth variety to encode the corner constraints, and obtain Gowers-norm control of the counting operator, bypassing Weyl differencing. A dimension-based analysis together with Lang–Weil bounds yields a robust geometric backbone, while a degree-lowering step (in the spirit of Peluse and Kuca) delivers a uniform power-saving error term of $O(p^{-1/40960})$, with the exponent remaining uniform across degrees of $P,Q$. The result extends quantitative Szemerédi-type statements to rational-function progressions in two dimensions, providing a new tool for understanding higher-dimensional polynomial-like configurations in finite fields and highlighting the role of algebraic geometry in additive combinatorics.

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 $\mathbb{F}_p^2$.

Theorems & Definitions (29)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Definition 3.1
• Theorem 3.2
• Proposition 3.3
• proof
• Remark 3.4
• Proposition 3.5