Erdős [Er46] asked whether every set of $n$ distinct points in $\mathbb{R}^2$ determines $\gg \frac{n}{\sqrt{\log n}}$ many distinct distances.

The square grid construction, going back to Erdős and Moser, shows that $\frac{n}{\sqrt{\log n}}$ is the correct order if the conjecture is true: there are configurations whose number of distinct distances is $O(\frac{n}{\sqrt{\log n}})$.