Sum-product theorems and incidence geometry

  • Mei-Chu Chang

    University of California, Riverside, United States
  • Jozsef Solymosi

    University of British Columbia, Vancouver, Canada


In this paper we prove the following theorems in incidence geometry. \proclaim{1} There is such that for any , and if there are many distinct lines between and for all , then 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 \delta 0 P_1, P_2, P_3 Q_1, \cdots, Q_n \in \Bbb C^2 \leq c n^{1/2} P_i Q_j i,j P\in \Bbb C^2\smallsetminus\{P_1,P_2,P_3\} \delta n P Q_j\( .\endproclaim \proclaim{3} Given \) c>0 \epsilon 0 P_1, P_2, P_3 Q_1, \cdots, Q_n \in \Bbb C^2 \Bbb R^2 \leq c n^{1/2} P_i Q_j i,j P L(P_1,P_2) n^{1-\epsilon} n/\log n P 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 \Bbb Q$.

Cite this article

Mei-Chu Chang, Jozsef Solymosi, Sum-product theorems and incidence geometry. J. Eur. Math. Soc. 9 (2007), no. 3, pp. 545–560

DOI 10.4171/JEMS/87