On the polynomial Szemer\'edi theorem in finite fields

TL;DR

This paper proves that large subsets of finite fields contain polynomial progressions defined by linearly independent polynomials, extending Szemerédi's theorem to polynomial configurations in finite fields.

Contribution

It establishes the existence of polynomial progressions in subsets of finite fields with size above a certain threshold, under large characteristic conditions.

Findings

Subsets of size at least q^{1-γ} contain polynomial progressions

Existence of polynomial progressions is guaranteed for large characteristic fields

Quantitative bounds depend on polynomial independence and field size

Abstract

Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $\gamma>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-\gamma}$ contains a nontrivial polynomial progression $x,x+P_1(y),\dots,x+P_m(y)$, provided the characteristic of $\mathbb{F}_q$ is large enough.