Distinct distance estimates and low degree polynomial partitioning

Abstract

We give a shorter proof of a slightly weaker version of a theorem of Nets Katz and the author. We prove that if a set of $L$ lines in $\mathbb{R}^3$ contains at most $L^{1/2}$ lines in any low degree algebraic surface, then the number of $r$-rich points is at most $C_\epsilon L^{(3/2) + \epsilon} r^{-2}$. Nets and I used this estimate to prove a distinct distance estimate for points in the plane. With the slightly weaker theorem in this paper, we get a slightly weaker distinct distance estimate: any set of $N$ points in $\mathbb{R}^2$ determines at least $c_\epsilon N^{1 - \epsilon}$ distinct distances.