Chabauty limits of Fermat spirals

A Fermat spiral is a set of points of the form $\sqrt{n}e^{2πiαn}$ for $α\in \mathbb{R}$. In this paper we prove that the Chabauty limits of Fermat spirals are always closed subgroups of $\mathbb{R}^2$, and conclude that no Fermat spirals are dense forests. Furthermore, we show that if $α$ is badly approximable the Chabauty limits are always lattices, for which we give a characterisation.