Spiral Delone sets in relative metric

TL;DR

This paper characterizes when a spiral lattice forms a Delone set in the relative metric, linking this property to the arithmetic nature of its rotation angle.

Contribution

It establishes a precise criterion connecting the Delone property of spiral lattices to the badly approximable nature of their rotation angles.

Findings

Spiral lattices are Delone sets in the relative metric if and only if their rotation angle is badly approximable.

Provides a characterization linking geometric properties of spiral sets to number-theoretic conditions.

Enhances understanding of the structure of spiral Delone sets in relation to their rotation angles.

Abstract

A general Archimedean spiral lattice is a Delone set in the relative distance if and only if its rotation angle is badly approximable.