An example related to the Erdos-Falconer question over arbitrary finite fields
Brendan Murphy, Giorgis Petridis·2019-05-13·via math.CO updates on arXiv.org
There exists an infinite family of examples of subsets of $\mathbb{F}_q^2$ with $q^{4/3}$ elements whose distance sets are not the whole of $\mathbb{F}_q$.