Sum-product theorems and incidence geometry

Abstract

In this paper we prove the following theorems in incidence geometry. \proclaim{1} There is $\delta >0$ such that for any $P\_1, \cdots, P\_4$, and $Q\_1, \cdots, Q\_n \in \Bbb C^2,$ if there are $\leq n^{\frac {1+\delta}2}$ 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 $then the cross ratio of the four points is algebraic.\endproclaim \proclaim{2} Given$ c>0$, there is$ \delta 0 $such that for any$ P\_1, P\_2, P\_3 $noncollinear, and$ Q\_1, \cdots, Q\_n \in \Bbb C^2$, if there are$ \leq c n^{1/2} $many distinct lines between$ P\_i $and$ Q\_j $for all$ i,j$, then for any$ P\in \Bbb C^2\smallsetminus\\{P\_1,P\_2,P\_3\\}$, we have$ \delta n $distinct lines between$ P $and$ Q\_j$.\endproclaim \proclaim{3} Given$ c>0$, there is$ \epsilon 0 $such that for any$ P\_1, P\_2, P\_3 $collinear, and$ Q\_1, \cdots, Q\_n \in \Bbb C^2 $(respectively,$ \Bbb R^2$), if there are$ \leq 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-\epsilon} $(resp.$ n/\log n$) distinct lines between$ P $and$ Q\_j$.\endproclaim The main ingredients used are the subspace theorem, Balog-Szemer'edi-Gowers Theorem, and Szemer'edi-Trotter Theorem. We also generalize the theorems to high dimensions, extend Theorem 1 to$ \Bbb F\_p^2$, and give the version of Theorem 2 over$ \Bbb Q$.