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**Higher dimensional spiral Delone sets.**
*(English)*
Zbl 1508.11074

Summary: 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 \(\{ \sqrt[n]{k}\cdot\boldsymbol{u}_k\}_{k\geqslant 1}\), where \((\boldsymbol{u}_k)_{k\geqslant 1}\) 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 \((\boldsymbol{u}_k)_{k\geqslant 1}\). This allows for the construction of explicit examples of spiral Delone sets in \(\mathbb{R}^n\) for all \(n\geqslant 2\), which boils down to finding a sequence of points in \(\mathbb{S}^{n-1}\) enjoying some optimal distribution properties.

Define a spiral set in \(\mathbb{R}^n\) as a set of points of the form \(\{ \sqrt[n]{k}\cdot\boldsymbol{u}_k\}_{k\geqslant 1}\), where \((\boldsymbol{u}_k)_{k\geqslant 1}\) 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 \((\boldsymbol{u}_k)_{k\geqslant 1}\). This allows for the construction of explicit examples of spiral Delone sets in \(\mathbb{R}^n\) for all \(n\geqslant 2\), which boils down to finding a sequence of points in \(\mathbb{S}^{n-1}\) enjoying some optimal distribution properties.

### MSC:

11J71 | Distribution modulo one |

37A44 | Relations between ergodic theory and number theory |

92C15 | Developmental biology, pattern formation |

92C80 | Plant biology |

11B05 | Density, gaps, topology |

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\textit{F. Adiceam} and \textit{I. Tsokanos}, Funct. Approximatio, Comment. Math. 67, No. 1, 21--46 (2022; Zbl 1508.11074)

### References:

[1] | S. Akiyama, Spiral Delone sets and three distance theorem, to appear in Nonlinearity, Nonlinearity 33 (2020), no. 5, 2533-2540. · Zbl 1487.37050 |

[2] | J.S. Athreya, D. Aulicino, and W.P. Hooper, Platonic solids and high genus covers of lattice surfaces, submitted, available at https://arxiv.org/abs/1811.04131. · Zbl 1502.14072 |

[3] | W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers, Math. Ann. 296(4) (1993), 625-635. · Zbl 0786.11035 |

[4] | J. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics. No. 45, Cambridge: at the University Press, 1957. · Zbl 0077.04801 |

[5] | M. Drmota, and R. Tichy, Sequences, discrepancies and applications, Lecture Notes in Mathematics, 1651, Springer-Verlag, Berlin, 1997. · Zbl 0877.11043 |

[6] | J. Marklof, Delone sets generated by square roots, to appear in Amer. Math. Monthly, available at https://arxiv.org/pdf/2003.08319.pdf. |

[7] | M. Ritoré, and E. Vernadakis, Isoperimetric inequalities in Euclidean convex bodies, Trans. Amer. Math. Soc. 367 (2015), no. 7, 4983-5014. · Zbl 1316.49052 |

[8] | W. Schmidt, W. Diophantine approximation, Lecture Notes in Mathematics 785, Berlin-Heidelberg-New York: Springer-Verlag, x, 299 p., 1980. · Zbl 0421.10019 |

[9] | V.A. Yudin, Placement of points on a torus and in a plane (Russian), Function theory, Trudy Inst. Mat. i Mekh. UrO RAN 11 (2005), no. 2, 196-200; Proc. Steklov Inst. Math. (Suppl.) 2005no., suppl. 2, S211-S216. · Zbl 1143.52303 |

[10] | Is there a generalisation of the “sunflower spiral” to higher dimensions?, MathOverFlow discussion 24850, available at https://mathoverflow.net. |

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