Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Higher Dimensional Spiral Delone Sets   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 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 $\mathbb{R}^n$ as a set of points of the form
$\left\{\sqrt[n]{k}\cdot\boldsymbol{u}_k\right\}_{k\ge 1}$, where
$\left(\boldsymbol{u}_k\right)_{k\ge 1}$ 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 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 $\left(\boldsymbol{u}_k\right)_{k\ge 1}$. 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.

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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: cs
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio