Few distinct distances implies no heavy lines or circles

TL;DR

This paper investigates the structure of planar point sets with few distinct distances, showing such sets cannot have heavily populated lines or circles, using advanced combinatorial and algebraic geometry techniques.

Contribution

It introduces a novel approach combining bipartite Elekes-Sharir framework with additive combinatorics and algebraic geometry to analyze point sets with few distances.

Findings

No line contains  n^{7/8} points of P.

No circle contains  n^{5/6} points of P.

Sets with few distances are structurally constrained.

Abstract

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distances, then no line contains \Omega(n^{7/8}) points of P and no circle contains \Omega(n^{5/6}) points of P. We rely on the bipartite and partial variant of the Elekes-Sharir framework that was presented by Sharir, Sheffer, and Solymosi in \cite{SSS13}. For the case of lines we combine this framework with a theorem from additive combinatorics, and 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 \cite{WYZ13}. A significant difference between our approach and that of \cite{SSS13} (and other recent extensions) 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.