Distinct Distance Estimates and Low Degree Polynomial Partitioning

We give a shorter proof of a slightly weaker version of a theorem fromGuth and Katz (Ann Math 181:155–190, 2015): we prove that if L is a set of L lines in R3 with at most L1/2 lines in any low degree algebraic surface, then the number of r -rich points of L is L(3/2)+εr−2. This result is one of the main ingredients in the proof of the distinct distance estimate in Guth and Katz (2015). With our slightly weaker theorem, we get a slightly weaker distinct distance estimate: any set of N points inR2 determines at least cεN 1−ε distinct distances.