Higher Dimensional Spiral Delone Sets
Faustin Adiceam, Ioannis Tsokanos

TL;DR
This paper characterizes when higher-dimensional spiral sets are Delone, linking their properties to spherical sequence distributions, and constructs explicit examples in all dimensions, extending known planar results.
Contribution
It extends the theory of spiral Delone sets from the plane to higher dimensions by providing a characterization and explicit constructions.
Findings
Characterization of spiral Delone sets via packing and covering conditions.
Explicit construction of spiral Delone sets in all dimensions.
Extension of planar spiral set theory to higher dimensions.
Abstract
A Delone set in is a set such that (a) the distance between any two of its points is uniformly bounded below by a strictly positive constant and such that (b) the distance from any point to the remaining points in the set is uniformly bounded above. Delone sets are thus sets of points enjoying nice spacing properties, and appear therefore naturally in mathematical models for quasicrystals. Define a spiral set in as a set of points of the form , where is a sequence in the unit sphere . In the planar case , spiral sets serve as natural theoretical models in phyllotaxis (the study of configurations of leaves on a plant stem), and an important example in this class includes the sunflower spiral. Recent works by Akiyama, Marklof and Yudin…
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Taxonomy
TopicsQuasicrystal Structures and Properties
