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

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 However, a detailed proof remains unpublished for many years.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} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3867969} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
3. M. Bouniaev, N. Dolbilin, “The local theory for regular systems in the context of $t$-bonded sets”, Symmetry-Basel, 10:5 (2018), 159
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
5. N. Dolbilin, M. Bouniaev, “Regular t-bonded systems in R-3”, Eur. J. Comb., 80 (2019), 89–101
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