AI 中文总结
研究有限域上多项式配置的同时流行差现象，证明存在更强现象成立，但也有严格限制，即固定素数\(p\)、\(n\)趋于无穷时，对某些集合\(A\subseteq\mathbb F_p^n\)，要求\(d\)和\(2d\)同时为三项等差数列流行差不成立。
AI 中文摘要
格林流行差定理表明，对于任意\(\varepsilon>0\)，所有足够大的素数\(p\)，以及每个密度为\(\alpha\)的集合\(A\subseteq\mathbb F_p\)，存在非零\(d\in\mathbb F_p\)使得\(\mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq \alpha^3-\varepsilon\)。我们证明对于多项式配置存在更强的同时流行差现象。即若\(\mathcal P=\{P_1,\dots,P_k\} \subset \mathbb Z[t]\)是固定的零常数项线性无关多项式集合，对于任意\(\varepsilon>0\)，所有足够大的素数\(p\)，以及每个密度为\(\alpha\)的集合\(A\subseteq\mathbb F_p\)，存在非零\(d\in\mathbb F_p\)使得\(\mathbb E_{x\in\mathbb F_p} 1_A(x) \prod_{i=1}^k 1_A\bigl(x+P_i(d)\bigr)^{\omega_i} \geq \alpha^{1+\sum_i\omega_i}-\varepsilon\)对每个\(\omega=(\omega_1,\dots,\omega_k)\in\{0,1\}^k\)同时成立。我们还证明这种同时流行差现象有严格限制，对于每个足够大的素数\(p\)，存在常数\(c>0\)，对于所有足够大的\(n\)，能找到密度为\(1/2+o_n(1)\)的集合\(A\subseteq\mathbb F_p^n\)满足\(\max_{d\neq 0} \min\left\{ \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+d)1_A(x+2d), \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+2d)1_A(x+4d) \right\} \leq \frac18-c\)。
英文摘要
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.
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