  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.
