# Sum-product theorems and incidence geometry

• Volume: 009, Issue: 3, page 545-560
• ISSN: 1435-9855

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## Abstract

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In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any ${P}_{1},\cdots ,{P}_{4}$, and ${Q}_{1},\cdots ,{Q}_{n}\in {ℂ}^{2}$, if there are $\le {n}^{\left(1+\delta \right)/2}$ 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. 2. Given $c>0$, there is $\delta >0$ such that for any ${P}_{1},{P}_{2},{P}_{3}\in {ℂ}^{2}$ noncollinear, and ${Q}_{1},\cdots ,{Q}_{n}\in {ℂ}^{2}$, if there are $\le c{n}^{1/2}$ many distinct lines between ${P}_{i}$ and ${Q}_{j}$ for all $i$, $j$, then for any $P\in {ℂ}^{2}\setminus \left\{{P}_{1},{P}_{2},{P}_{3}\right\}$, we have $\delta n$ distinct lines between $P$ and ${Q}_{j}$. 3. Given $c>0$, there is $ϵ>0$ such that for any ${P}_{1},{P}_{2},{P}_{3}\in {ℂ}^{2}$ collinear, and ${Q}_{1},\cdots ,{Q}_{n}\in {ℂ}^{2}$ (respectively, ${ℝ}^{2}$), if there are $\le c{n}^{1/2}$ many distinct lines between ${P}_{i}$ and ${Q}_{j}$ for all $i,j$, then for any $P$ not lying on the line $L\left({P}_{1},{P}_{2}\right)$, we have at least ${n}^{1-ϵ}$ (resp. $n/logn$) 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 ${𝔽}_{p}^{2}$, and give the version of Theorem 2 over $ℚ$.

## How to cite

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Chang, Mei-Chu, and Solymosi, Jozsef. "Sum-product theorems and incidence geometry." Journal of the European Mathematical Society 009.3 (2007): 545-560. <http://eudml.org/doc/277465>.

@article{Chang2007,
abstract = {In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\dots ,P_4$, and $Q_1,\dots ,Q_n\in \mathbb \{C\}^2$, if there are $\le n^\{(1+\delta )/2\}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\dots ,P_4$ are collinear. If the number of the distinct lines is $<cn^\{1/2\}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in \mathbb \{C\}^2$ noncollinear, and $Q_1,\dots ,Q_n\in \mathbb \{C\}^2$, if there are $\le cn^\{1/2\}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in \mathbb \{C\}^2\setminus \lbrace P_1,P_2,P_3\rbrace$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given $c>0$, there is $\epsilon >0$ such that for any $P_1,P_2,P_3\in \mathbb \{C\}^2$ collinear, and $Q_1, \dots ,Q_n \in \mathbb \{C\}^2$ (respectively, $\mathbb \{R\}^2$), if there are $\le c n^\{1/2\}$ 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 $Q_j$. 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 $\mathbb \{F\}_p^2$, and give the version of Theorem 2 over $\mathbb \{Q\}$.},
author = {Chang, Mei-Chu, Solymosi, Jozsef},
journal = {Journal of the European Mathematical Society},
language = {eng},
number = {3},
pages = {545-560},
publisher = {European Mathematical Society Publishing House},
title = {Sum-product theorems and incidence geometry},
url = {http://eudml.org/doc/277465},
volume = {009},
year = {2007},
}

TY - JOUR
AU - Chang, Mei-Chu
AU - Solymosi, Jozsef
TI - Sum-product theorems and incidence geometry
JO - Journal of the European Mathematical Society
PY - 2007
PB - European Mathematical Society Publishing House
VL - 009
IS - 3
SP - 545
EP - 560
AB - In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\dots ,P_4$, and $Q_1,\dots ,Q_n\in \mathbb {C}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\dots ,P_4$ are collinear. If the number of the distinct lines is $<cn^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in \mathbb {C}^2$ noncollinear, and $Q_1,\dots ,Q_n\in \mathbb {C}^2$, if there are $\le cn^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in \mathbb {C}^2\setminus \lbrace P_1,P_2,P_3\rbrace$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given $c>0$, there is $\epsilon >0$ such that for any $P_1,P_2,P_3\in \mathbb {C}^2$ collinear, and $Q_1, \dots ,Q_n \in \mathbb {C}^2$ (respectively, $\mathbb {R}^2$), if there are $\le c n^{1/2}$ 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 $Q_j$. 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 $\mathbb {F}_p^2$, and give the version of Theorem 2 over $\mathbb {Q}$.
LA - eng
UR - http://eudml.org/doc/277465
ER -

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