Sum-product theorems and incidence geometry
- Autores: Mei-Chu Chang, Jozsef Solymosi
- Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 9, Nº 3, 2007, págs. 545-560
- Idioma: inglés
- DOI: 10.4171/jems/87
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Resumen
In this paper we prove the following theorems in incidence geometry. \proclaim{1} There is $\delta >0$ such that for any $P_1, \cdots, P_4$, and $Q_1, \cdots, Q_n \in \Bbb C^2,$ if there are $\leq n^{\frac {1+\delta}2}$ 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.\endproclaim \proclaim{2} Given $c>0$, there is $\delta 0$ such that for any $P_1, P_2, P_3$ noncollinear, and $Q_1, \cdots, Q_n \in \Bbb C^2$, if there are $\leq c n^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i,j$, then for any $P\in \Bbb C^2\smallsetminus\{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$.\endproclaim \proclaim{3} Given $c>0$, there is $\epsilon 0$ such that for any $P_1, P_2, P_3$ collinear, and $Q_1, \cdots, Q_n \in \Bbb C^2$ (respectively, $\Bbb R^2$), if there are $\leq c n^{1/2}$ 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 $Q_j$.\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 $\Bbb F_p^2$, and give the version of Theorem 2 over $\Bbb Q$.
