On the polynomial Szemerédi theorem in finite fields

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 $\gamma >0$ 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 that the characteristic of ${\mathbb{F}}_q$ is large enough.