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Fundam. Prikl. Mat., 2016, Volume 21, Issue 6, Pages 115–141 (Mi fpm1771)  

This article is cited in 4 scientific papers (total in 4 papers)

Delone sets in $\mathbb{R}^3$: regularity conditions

N. P. Dolbilin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called $10R$-theorem. This theorem asserts that identity of neighborhoods within a radius $10R$ of all points of a Delone set (in other words, an $(r,R)$-system) in $\mathrm{3D}$ Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the $10R$-theorem. In the proof, we use some recent results of the author, which simplify the proof.

Funding Agency Grant Number
Russian Science Foundation 14-11-00414


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UDC: 514.15+514.17+514.8+548.1

Citation: N. P. Dolbilin, “Delone sets in $\mathbb{R}^3$: regularity conditions”, Fundam. Prikl. Mat., 21:6 (2016), 115–141

Citation in format AMSBIB
\Bibitem{Dol16}
\by N.~P.~Dolbilin
\paper Delone sets in $\mathbb{R}^3$: regularity conditions
\jour Fundam. Prikl. Mat.
\yr 2016
\vol 21
\issue 6
\pages 115--141
\mathnet{http://mi.mathnet.ru/fpm1771}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. P. Dolbilin, “Delone sets in $\mathbb R^3$ with $2R$-regularity conditions”, Proc. Steklov Inst. Math., 302 (2018), 161–185  mathnet  crossref  crossref  isi  elib
    2. I. A. Baburin, M. Bouniaev, N. Dolbilin, N. Yu. Erokhovets, A. Garber, S. V. Krivovichev, E. Schulte, “On the origin of crystallinity: a lower bound for the regularity radius of Delone sets”, Acta Crystallogr. Sect. A, 74:6 (2018), 616–629  crossref  mathscinet  isi  scopus
    3. M. Bouniaev, N. Dolbilin, “The local theory for regular systems in the context of $t$-bonded sets”, Symmetry-Basel, 10:5 (2018), 159  crossref  isi  scopus
    4. N. Dolbilin, “Delone sets: local identity and global symmetry”, Discrete Geometry and Symmetry, Springer Proceedings in Mathematics & Statistics, Dedicated to Karoly Bezdek and Egon Schulte on the Occasion of Their 60Th Birthdays, 234, eds. M. Conder, A. Deza, A. Weiss, Springer, 2018, 109–125  crossref  mathscinet  zmath  isi  scopus
  • Фундаментальная и прикладная математика
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