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 ${\mathbb{F}}^2$. 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 $R^{\ast }(2\to \alpha )$ holds for $\alpha >\frac{10}{3}$ in ${\mathbb{F}}^3$ when $p\equiv 3\mathrm{mod}4$.

arXiv commentMinor revision. 17 pages, 0 figures. After completing this paper, we learned that Prof. Lewko had independently obtained similar results. We are grateful to Prof. Lewko for sharing his manuscript and valuable suggestions