Bourgain's incidence set problem -

Abstract

This thesis investigates Bourgain's incidence set problem. This problem concerns the intersection patterns of lines in space. More precisely, given a set X in R3, and a set L of N2 lines in R3 satisfying the following conditions: (i)Each line of L contains at least N points of X. (ii)No more than N lines of L can lie in the same plane, We need to nd a lower bound on the cardinality of X. Bourgain conjectured in 2004 that the cardinality of X is bounded from below by C N3, where C is an absolute constant. This conjecture remained open until 2010, when it was resolved by Guth and Katz using polynomial partitioning methods. In Chapter 1, we present some of the results that were known before the 2010 paper of Guth and Katz. These results are obtained by using standard counting methods, and they fall short of the conjectured estimate. In Chapter 2, we present the background needed for using the polynomial partition-ing method. In Chapter 3, we study flat points. In Chapter 4 we present the paper in which Guth and Katz prove the conjecture. In Chapter 5, we give a proof of the conjecture by using theSzeméredi- Trotter theorem in R3 following the development in Guth's book. In Chapter 6, we investigate the same problem but in the finite field setting following the work of Ellenberg and Hablicsek.