Simultaneous popular polynomial differences over finite fields

Abstract

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 \(α\), there exists a nonzero \(d\in\mathbb F_p\) such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq α^3-\varepsilon. \] 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 \(α\), there exists a nonzero \(d\in\mathbb F_p\) such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x) \prod_{i=1}^k 1_A\bigl(x+P_i(d)\bigr)^{ω_i} \geq α^{1+\sum_iω_i}-\varepsilon \] simultaneously for every \(ω=(ω_1,\dots,ω_k)\in\{0,1\}^k\). 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.