Sum-product theorems and incidence geometry

Abstract

In this paper we prove the following theorems in incidence geometry.

1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {\mathbb{C}}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i,j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$, then the cross ratio of the four points is algebraic.
2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3$ noncollinear, and $Q_1,\cdots ,Q_n\in {\mathbb{C}}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i,j$, then for any $P\in {\mathbb{C}}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$
3. Given $c>0$, there is $ϵ>0$ such that for any $P_1,P_2,P_3$ collinear, and $Q_1,\cdots ,Q_n\in {\mathbb{C}}^2$ (respectively, ${\mathbb{R}}^2$), if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i,j$, then for any $P$ not lying on the line $L(P_1,P_2)$, we have at least $n^{1-ϵ}$(resp. $n/\log n$) distinct lines between $P$ and $Q_j$.

The main ingredients used are the subspace theorem, Balog–Szemerédi–Gowers Theorem, and Szemerédi–Trotter Theorem. We also generalize the theorems to high dimensions, extend Theorem 1 to ${\mathbb{F}}_p^2$, and give the version of Theorem 2 over $\mathbb{Q}$.