# Sum-product theorems and incidence geometry

Mei-Chu Chang; Jozsef Solymosi

Journal of the European Mathematical Society (2007)

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

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topChang, 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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