Distinct Distance Estimates and Low Degree Polynomial Partitioning

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 $$\mathfrak {L}$$L is a set of $$L$$L lines in $$\mathbb {R}^3$$R3 with at most $$L^{1/2}$$L1/2 lines in any low degree algebraic surface, then the number of $$r$$r-rich points of $$\mathfrak {L}$$L is $$\lesssim L^{(3/2) + \varepsilon } r^{-2}$$≲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 $$N$$N points in $$\mathbb {R}^2$$R2 determines at least $$c_{\varepsilon } N^{1 -{\varepsilon }}$$cεN1-ε distinct distances.