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 $Q_1,\cdots ,Q_n\in {\mathbb{C}}^2,$ if there are $\le n^{\frac{1+\delta }{2}}$ 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$,thereis$ \delta 0 $suchthatforany$ P_1, P_2, P_3 $noncollinear,and$ Q_1, \cdots, Q_n \in \Bbb C^2$,ifthereare$ \leq c n^{1/2} $manydistinctlinesbetween$ P_i $and$ Q_j $forall$ i,j$,thenforany$ P\in \Bbb C^2\smallsetminus\{P_1,P_2,P_3\}$,wehave$ \delta n $distinctlinesbetween$ P $and$ Q_j\( .\endproclaim \proclaim{3} Given \) c>0$,thereis$ \epsilon 0 $suchthatforany$ P_1, P_2, P_3 $collinear,and$ Q_1, \cdots, Q_n \in \Bbb C^2 $(respectively,$ \Bbb R^2$),ifthereare$ \leq c n^{1/2} $manydistinctlinesbetween$ P_i $and$ Q_j $forall$ i,j$,thenforany$ P $notlyingontheline$ L(P_1,P_2)$,wehaveatleast$ n^{1-\epsilon} $(resp.$ n/\log n$)distinctlinesbetween$ 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$,andgivetheversionofTheorem2over$ \Bbb Q$.