The Szemerédi-Trotter Estimate in Finite Field with its Applications

Abstract: We prove a sharp Szemerédi-Trotter estimate [\mathcal{I}(A,\mathcal{L})\lesssim \frac{|A||\mathcal{L}|}{p}+|A|^{2/3}|\mathcal{L}|^{2/3}+|A|+|\mathcal{L}|] for prime finite field $\mathbb{F}=\mathbb{F}_p$ by a new polynomial decomposition theorem. As applications, we first prove the sharp Furstenberg set estimate in $\mathbb{F}<sup>2$. Secondly, we improve sum-product estimate [\max{|A+A|,|A\cdot A|}\gtrsim\min{(p|A|)^{1/2},|A|^{5/4}},\quad A\subset\mathbb{F}.] Finally, we improve the Fourier restriction estimate $R<sup>*(2\toα)$ holds for $α&gt;\frac{10}{3}$ in $\mathbb{F}<sup>3$ when $p\equiv 3\mod 4$.