Simultaneous popular polynomial differences over finite fields

Green's popular difference theorem says that for every $\varepsilon>0$, all sufficiently large primes $p$, and every set $A\subseteq\mathbb F_p$ of density $\alpha$, there exists a nonzero $d\in\mathbb F_p$ such that $$\mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq \alpha^3-\varepsilon.$$ We show that a stronger simultaneous popular difference phenomenon holds for polynomial configurations. Namely, if $\mathcal P=\{P_1,\dots,P_k\} \subset \mathbb Z[t]$ is a fixed collection of linearly independent polynomials with zero constant terms, we show that for every $\varepsilon>0$, all sufficiently large primes $p$, and every set $A\subseteq\mathbb F_p$ of density $\alpha$, there exists a nonzero $d\in\mathbb F_p$ such that $$\mathbb E_{x\in\mathbb F_p} 1_A(x) \prod_{i=1}^k 1_A\bigl(x+P_i(d)\bigr)^{\omega_i} \geq \alpha^{1+\sum_i\omega_i}-\varepsilon$$ simultaneously for every $\omega=(\omega_1,\dots,\omega_k)\in\{0,1\}^k$. We also show that such simultaneous popular difference phenomena have sharp limitations by proving that for every sufficiently large prime $p$, there is a constant $c>0$ such that, for all sufficiently large $n$, one can find a set $A\subseteq\mathbb F_p^n$ of density $1/2+o_n(1)$ satisfying $$\max_{d\neq 0} \min\left\{ \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+d)1_A(x+2d), \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+2d)1_A(x+4d) \right\} \leq \frac18-c.$$ That is, the strengthening of Green's result, in this case over $\mathbb F_p^n$ for $p$ fixed and $n$ tending to infinity, requiring that both $d$ and $2d$ are simultaneously popular differences for three-term arithmetic progressions is false.