2020/10/13 by Faustin Adiceam, Adiceam, Faustin, Ioannis Tsokanos +1
Materials Science · #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT) #Quasicrystal Structures and Properties
paper · doi:10.48550/arxiv.2010.06428
openalex publication_date 2020/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Delone set in ℝn 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 ℝn as a set of points of the form \√[n]k⋅\boldsymboluk\k≥ 1, where (\boldsymboluk)k≥ 1 is a sequence in the unit sphere \mathbbSn-1. In the planar case n=2, 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 provide a reasonable complete characterisation of planar spiral sets which are also Delone. A related problem that has emerged in several places in the literature over the past fews years is to determine whether this theory can be extended to higher dimensions, and in particular to show the existence of spiral Delone sets in any dimension. This paper addresses this question by characterising the Delone property of a spiral set in terms of packing and covering conditions satisfied by the spherical sequence (\boldsymboluk)k≥ 1. This allows for the construction of explicit examples of spiral Delone sets in ℝn for all n≥ 2, which boils down to finding a sequence of points in \mathbbSn-1 enjoying some optimal distribution properties.