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September 2022 Higher dimensional spiral Delone sets
Faustin Adiceam, Ioannis Tsokanos
Funct. Approx. Comment. Math. 67(1): 21-46 (September 2022). DOI: 10.7169/facm/1958

Abstract

A Delone set in $\mathbb{R}^n$ 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 in $\mathbb{R}^n$ to the set is uniformly bounded above.Delone sets are thus sets of points enjoying nice spacing properties, and they therefore appear naturally in mathematical models for quasicrystals. Define a spiral set in $\mathbb{R}^n$ as a set of points of the form $\{\sqrt[n]{k}\cdot\bm{u}_k\}_{k\ge 1}$, where $(\bm{u}_k)_{k\ge 1}$ is a sequence in the unit sphere $\mathbb{S}^{n-1}$. 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 few 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 $(\mathbf{u}_k)_{k\ge 1}$. This allows for the construction of explicit examples of spiral Delone sets in $\mathbb{R}^n$ for all $n\ge 2$, which boils down to finding a sequence of points in $\mathbb{S}^{n-1}$ enjoying some optimal distribution properties.

Citation

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Faustin Adiceam. Ioannis Tsokanos. "Higher dimensional spiral Delone sets." Funct. Approx. Comment. Math. 67 (1) 21 - 46, September 2022. https://doi.org/10.7169/facm/1958

Information

Published: September 2022
First available in Project Euclid: 13 April 2022

Digital Object Identifier: 10.7169/facm/1958

Subjects:
Primary: 11B05 , 92C15
Secondary: 52C23

Keywords: Delone sets , distribution modulo one , phyllotaxis , spiral sets

Rights: Copyright © 2022 Adam Mickiewicz University

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Vol.67 • No. 1 • September 2022
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