Chabauty Limits of Fermat Spirals

TL;DR

This paper investigates the limiting behavior of Fermat spirals under Chabauty topology, showing they form closed subgroups or lattices depending on the approximation properties of b5, and establishing that they are not dense forests.

Contribution

It characterizes the Chabauty limits of Fermat spirals as closed subgroups and lattices, providing a new understanding of their geometric and algebraic structure.

Findings

Chabauty limits of Fermat spirals are always closed subgroups of a0b5^2.

If b5 is badly approximable, the limits are lattices.

Fermat spirals are not dense forests.

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.