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 Trudy Mat. Inst. Steklova, 2018, Volume 302, Pages 176–201 (Mi tm3936)

Delone sets in $\mathbb R^3$ with $2R$-regularity conditions

N. P. Dolbilin

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: A regular system is the orbit of a point with respect to a crystallographic group. The central problem of the local theory of regular systems is to determine the value of the regularity radius, i.e., the radius of neighborhoods/clusters whose identity in a Delone $(r,R)$‑set guarantees its regularity. In this paper, conditions are described under which the regularity of a Delone set in three-dimensional Euclidean space follows from the pairwise congruence of small clusters of radius $2R$. Combined with the analysis of one particular case, this result also implies the proof of the “$10R$-theorem,” which states that the congruence of clusters of radius $10R$ in a Delone set implies the regularity of this set.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968518030081

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 302, 161–185

Bibliographic databases:

UDC: 514.1+514.8+548.1

Citation: N. P. Dolbilin, “Delone sets in $\mathbb R^3$ with $2R$-regularity conditions”, Topology and physics, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 302, MAIK Nauka/Interperiodica, Moscow, 2018, 176–201; Proc. Steklov Inst. Math., 302 (2018), 161–185

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3936
• https://doi.org/10.1134/S0371968518030081
• http://mi.mathnet.ru/eng/tm/v302/p176

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This publication is cited in the following articles:
1. N. P. Dolbilin, M. I. Shtogrin, “Lokalnye gruppy v mnozhestvakh Delone: gipoteza i rezultaty”, UMN, 76:6(462) (2021), 193–194
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