1. Sum-product theorems and incidence geometry.
- Author
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Mei-Chu Chang and Solymosi, József
- Subjects
- *
SUBSPACES (Mathematics) , *GENERALIZATION , *COMBINATORICS , *MATRICES (Mathematics) , *CAUCHY problem - Abstract
We prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P1, . . ., P4 ∈ C², and Q1, . . .,Qn ∈ C², if there are C² n(1+δ)/2 distinct lines between Pi and Qj for all i, j, then P1, . . ., P4 are collinear. If the number of the distinct lines is < cn1/2, then the cross ratio of the four points is algebraic. 2. Given c > 0, there is δ > 0 such that for any P1, P2, P3 ∈ C² noncollinear, and Q1, . . .,Qn ∈ C², if there are ≤ cn1/2 distinct lines between Pi and Qj for all i, j, then for any P ∈ C² r {P1, P2, P3}, we have δn distinct lines between P and Qj. 3. Given c > 0, there is ∊ > 0 such that for any P1, P2, P3 ∈ C² (respectively, R²) collinear, and Q1, . . .,Qn ∈ C² (respectively, R²), if there are ∊ cn1/2 distinct lines between Pi and Qj for all i, j, then for any P not lying on the line L(P1, P2), we have at least n1-∊ (resp. n/log n) distinct lines between P and Qj. The main ingredients used are the subspace theorem, Balog-Szemerédi-Gowers theorem, and Szemer 'edi-Trotter theorem. We also generalize the theorems to higher dimensions, extend Theorem 1 to F²p, and give the version of Theorem 2 over Q. [ABSTRACT FROM AUTHOR]
- Published
- 2007
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