JournalsjemsVol. 9 , No. 3DOI 10.4171/jems/87

Sum-product theorems and incidence geometry

  • Mei-Chu Chang

    University of California, Riverside, United States
  • Jozsef Solymosi

    University of British Columbia, Vancouver, Canada
Sum-product theorems and incidence geometry cover

Abstract

In this paper we prove the following theorems in incidence geometry. \proclaim{1} There is δ>0\delta >0 such that for any P1,,P4P_1, \cdots, P_4, and Q1,,QnC2,Q_1, \cdots, Q_n \in \Bbb C^2, if there are n1+δ2\leq n^{\frac {1+\delta}2} many distinct lines between PiP_i and QjQ_j for all i,ji,j, then P1,,P4P_1, \cdots, P_4 are collinear. If the number of the distinct lines is thenthecrossratioofthefourpointsisalgebraic.\endproclaim\proclaim2Giventhen the cross ratio of the four points is algebraic.\endproclaim \proclaim{2} Given c>0,thereis, there is \delta 0 suchthatforanysuch that for any P_1, P_2, P_3 noncollinear,andnoncollinear, and Q_1, \cdots, Q_n \in \Bbb C^2,ifthereare, if there are \leq c n^{1/2} manydistinctlinesbetweenmany distinct lines between P_i andand Q_j forallfor all i,j,thenforany, then for any P\in \Bbb C^2\smallsetminus\{P_1,P_2,P_3\},wehave, we have \delta n distinctlinesbetweendistinct lines between P andand Q_j.\endproclaim\proclaim3Given.\endproclaim \proclaim{3} Given c>0,thereis, there is \epsilon 0 suchthatforanysuch that for any P_1, P_2, P_3 collinear,andcollinear, and Q_1, \cdots, Q_n \in \Bbb C^2 (respectively,(respectively, \Bbb R^2),ifthereare), if there are \leq c n^{1/2} manydistinctlinesbetweenmany distinct lines between P_i andand Q_j forallfor all i,j,thenforany, then for any P notlyingonthelinenot lying on the line L(P_1,P_2),wehaveatleast, we have at least n^{1-\epsilon} (resp.(resp. n/\log n)distinctlinesbetween) distinct lines between P andand Q_j.\endproclaimThemainingredientsusedarethesubspacetheorem,BalogSzemereˊdiGowersTheorem,andSzemereˊdiTrotterTheorem.Wealsogeneralizethetheoremstohighdimensions,extendTheorem1to.\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, and give the version of Theorem 2 over \Bbb Q$.