On sets with few distinct distances

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 $O(|A|<sup>2)$ distinct distances, then $|A-A|=O\left(|A|<sup>{2-\frac{2}{11}}\right)$. 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 $P \subset \mathbb R<sup>2$ of $N$ points determines $O(N/\sqrt {\log N})$ distinct distances, then there exists a reflection $\mathcal R$ and a set $P&#39; \subset P$ with $|P&#39;| =\Omega ( \log^{3/2} N)$ such that $\mathcal R(P&#39;) \subset P$. In other words, sets with few distinct distances have some degree of reflexive symmetry.