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 $\mathfrak {L}$ is a set of $L$ lines in $\mathbb {R}^3$ with at most $L^{1/2}$ lines in any low degree algebraic surface, then the number of $r$-rich points of $\mathfrak {L}$ is $\lesssim L^{(3/2) + \varepsilon } r^{-2}$. 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 $\mathbb {R}^2$ determines at least $c_{\varepsilon } N^{1 -{\varepsilon }}$ distinct distances.