Geodesic
Delone sets in $\mathbb{R}^3$: regularity conditions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 6, pp. 115-141.

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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 However, a detailed proof remains unpublished for many years.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. 
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N. P. Dolbilin. Delone sets in $\mathbb{R}^3$: regularity conditions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 6, pp. 115-141. http://geodesic.mathdoc.fr/item/FPM_2016_21_6_a4/

[1] Delone B. N., Dolbilin N. P., Shtogrin M. I., Galiulin R. V., “Lokalnyi kriterii pravilnosti sistemy tochek”, DAN SSSR, 227:1 (1976), 19–21 | MR | Zbl

[2] Dolbilin N. P., “Kriterii kristalla i lokalno antipodalnye mnozhestva Delone”, Vestnik ChelGU, 2015, no. 3 (358), 6–17

[3] Dolbilin N. P., Magazinov A. N., “Lokalno antipodalnye mnozhestva Delone”, UMN, 70:5 (425) (2015), 179–180 | DOI | MR | Zbl

[4] Dolbilin N. P., Magazinov A. N., “Teorema edinstvennosti dlya lokalno antipodalnykh mnozhestv Delone”, Tr. MIAN, 294, 2016, 230–236 | Zbl

[5] Feinman R., Leiton R., Sends M., Feinmanovskie lektsii po fizike, 7, Editorial URSS, M., 2004

[6] Shtogrin M., “Ob ogranichenii poryadka osi pauchka v lokalno pravilnoi sisteme Delone”, Geometry, Topology, Algebra and Number Theory, Applications, The International Conference dedicated to the 120-th anniversary of Boris Nikolaevich Delone (1890–1980), Tezisy doklada (Moscow, August 16–20, 2010), Steklov Math. Inst., M., 2010, 168–169

[7] Dolbilin N., “Delone sets with congruent clusters”, Struct. Chem., 26:6 (2016), 1725–1732 | DOI

[8] Dolbilin N., “Delone sets: local identity and global order”, Discrete Geometry and Symmetry, Dedicated to Károly Bezdek and Egon Schulte on the Occasion of Their 60th Birthdays, Springer, Berlin, 2017, 109–125, arXiv: 1608.06842