Distinct Distance Estimates and Low Degree Polynomial Partitioning

Abstract

We give a shorter proof of a slightly weaker version of a theorem from Guth and Katz (Ann Math 181:155–190, 2015): we prove that if L is a set of L lines in R[superscript 3] with at most L[superscript 1/2] lines in any low degree algebraic surface, then the number of r-rich points of is L is ≲ L[superscript (3/2) + ε] r[superscript -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 in R[superscript 2] c[subscript ε]N[superscript 1-ε] distinct distances.