Sum-product theorems and incidence geometry
Mei-Chu Chang, Jozsef Solymosi (2007)
Journal of the European Mathematical Society
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In this paper we prove the following theorems in incidence geometry. 1. There is such that for any , and , if there are many distinct lines between and for all , , then are collinear. If the number of the distinct lines is then the cross ratio of the four points is algebraic. 2. Given , there is such that for any noncollinear, and , if there are many distinct lines between and for all , , then for any , we have distinct lines between and . 3. Given...