# Sum-product theorems and incidence geometry

### Mei-Chu Chang

University of California, Riverside, United States### Jozsef Solymosi

University of British Columbia, Vancouver, Canada

## Abstract

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

- There is $δ>0$ such that for any $P_{1},⋯,P_{4}$, and $Q_{1},⋯,Q_{n}∈C_{2}$, if there are $≤n_{(1+δ)/2}$ many distinct lines between $P_{i}$ and $Q_{j}$ for all $i,j$, then $P_{1},⋯,P_{4}$ are collinear. If the number of the distinct lines is $<cn_{1/2}$, then the cross ratio of the four points is algebraic.
- Given $c>0$, there is $δ>0$ such that for any $P_{1},P_{2},P_{3}$ noncollinear, and $Q_{1},⋯,Q_{n}∈C_{2}$, if there are $≤cn_{1/2}$ many distinct lines between $P_{i}$ and $Q_{j}$ for all $i,j$, then for any $P∈C_{2}∖{P_{1},P_{2},P_{3}}$, we have $δn$ distinct lines between $P$ and $Q_{j}$
- Given $c>0$, there is $ϵ>0$ such that for any $P_{1},P_{2},P_{3}$ collinear, and $Q_{1},⋯,Q_{n}∈C_{2}$ (respectively, $R_{2}$), if there are $≤cn_{1/2}$ 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. $n/gn$) 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 $F_{p}$, and give the version of Theorem 2 over $Q$.

## 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