Three-term polynomial progressions in subsets of finite fields

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 $x,x+P_1(y),x+P_2(y)$.

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