任意域中的Szemerédi-Trotter定理
A Szemerédi-Trotter Theorem in Arbitrary Fields
AI总结：
本文在任意特征域上证明Szemerédi-Trotter定理，给出点线关联数的上界，并应用于抛物面延拓估计、Bourgain提取器及sum-product估计的改进。
AI中文摘要：
设$k$为特征$p>0$的域。我们证明在$k^2$中，$m$个点和$n$条线确定$O((mn)^{2/3}+m+n+mn/p)$个关联。在特征为零时，最后一项被省略。证明使用多项式方法，且当$m=n$时，该界在素数域上是尖锐的。作为应用，在$p\equiv 3\pmod4$的素数域上，我们获得$F_p^3$中抛物面在$r>10/3$时的$L^2\to L^r$延拓估计，并证明Bourgain抛物面提取器能从任何最小熵率大于$3/8=0.375$的独立源中提取，误差指数级小。我们还改进了正特征中小集合的sum-product估计。
英文摘要：
Let $k$ be a field of characteristic $p\ge0$. We prove that $m$ points and $n$ lines in $k^2$ determine at most $3(mn)^{2/3}+m+n+2mn/p$ incidences, the last term being omitted in characteristic zero. Over the prime field $\mathbb{F}_p$ the coefficient of $mn/p$ can be replaced by $1$. The proof uses the polynomial method, and for $m=n$ the bound is sharp up to an absolute constant over prime fields. As applications, over prime fields in which $-1$ is not a square we obtain the $L^2\to L^r$ extension estimate for the paraboloid in $\mathbb{F}_p^3$ for $r>10/3$. Over every odd prime field, we show that a two-source extractor construction of Bourgain has exponentially small error at every min-entropy rate greater than $1/3$. We also improve sum-product estimates for small sets in positive characteristic and obtain projection and Furstenberg estimates over prime fields. The incidence inequalities with exact constants have been formalized in Lean.
