On sets with few distinct distances

TL;DR

This paper investigates the structure of point sets with few distinct distances in the plane, showing they exhibit additive structure or reflexive symmetry, thus supporting the belief they resemble lattice-like configurations.

Contribution

It improves bounds on the additive structure of Cartesian product point sets with few distances and demonstrates that such sets exhibit partial reflexive symmetry.

Findings

Cartesian product sets with few distances have small difference sets.

Sets with few distances contain large subsets symmetric under reflection.

Results support the idea that such sets are structurally similar to lattices.

Abstract

It is widely believed that point sets in the plane which determine few distinct distances must have some special structure. In particular, such sets are believed to be similar to a lattice. This note considers two different ways to quantify this idea. Firstly, improving on a result of Hanson (see arXiv:1607.03442), it is proven that if $P= A \times A$ with $A \subset \mathbb R$ and $P$ determines $O(|A|^2)$ distinct distances, then $|A-A|=O\left(|A|^{2-\frac{2}{11}}\right)$. This result gives further evidence that cartesian products which determine few distinct distances have some additive structure. Secondly, it is shown that if a set $P \subset \mathbb R^2$ of $N$ points determines $O(N/\sqrt {\log N})$ distinct distances, then there exists a reflection $\mathcal R$ and a set $P' \subset P$ with $|P'| =\Omega ( \log^{3/2} N)$ such that $\mathcal R(P') \subset P$. In other words, sets with few distinct distances have some degree of reflexive symmetry.