Chabauty Limits of Fermat Spirals

Abstract

A Fermat spiral is a set of points of the form $\sqrt{n}{e}^{2\pi i\alpha n}$ for $\alpha \in \mathbb{R}$. 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 $\alpha$ is badly approximable the Chabauty limits are always lattices, for which we give a characterisation.

References in corpus (3)

• Spiral Delone sets and three distance theorem
• Area convergence of Voronoi cells on spiral lattices
• Visibility Properties of Spiral Sets