Three-term polynomial progressions in subsets of finite fields

TL;DR

This paper proves that subsets of finite fields with sufficiently large density necessarily contain polynomial progressions defined by linearly independent polynomials, extending previous results on polynomial configurations.

Contribution

It establishes new density bounds ensuring the existence of polynomial progressions in finite fields for linearly independent polynomials, answering an open question.

Findings

Subsets of density > p^{-1/24} contain polynomial progressions

Extension of Bourgain and Chang's results to more general polynomial progressions

Provides explicit density thresholds for polynomial configurations

Abstract

Bourgain and Chang recently showed that any subset of $\mathbb{F}_p$ of density $\gg p^{-1/15}$ contains a nontrivial progression $x,x+y,x+y^2$. We answer a question of theirs by proving that if $P_1,P_2\in\mathbb{Z}[y]$ are linearly independent and satisfy $P_1(0)=P_2(0)=0$, then any subset of $\mathbb{F}_p$ of density $\gg_{P_1,P_2}p^{-1/24}$ contains a nontrivial polynomial progression $x,x+P_1(y),x+P_2(y)$.