Few distinct distances implies no heavy lines or circles

Abstract

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set $\mathcal{P}$ of n points determines o(n) distinct distances, then no line contains Ω(n ^7/8) points of $\mathcal{P}$ and no circle contains Ω(n ^5/6) points of $\mathcal{P}$.

We rely on the partial variant of the Elekes-Sharir framework that was introduced by Sharir, Sheffer, and Solymosi in [19] for bipartite distinct distance problems. To prove our bound for the case of lines we combine this framework with a theorem from additive combinatorics, and for our bound for the case of circles we combine it with some basic algebraic geometry and a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang [20].

A significant difference between our approach and that of [19] (and of other related results) is that instead of dealing with distances between two point sets that are restricted to one-dimensional curves, we consider distances between one set that is restricted to a curve and one set with no restrictions on it.