Geodesic
Delone sets in $\mathbb R^3$ with $2R$-regularity conditions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 176-201

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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.
Keywords: Delone set, crystallographic group, regular system, regularity radius, cluster. 
@article{TM_2018_302_a7,
     author = {N. P. Dolbilin},
     title = {Delone sets in $\mathbb R^3$ with $2R$-regularity conditions},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {176--201},
     publisher = {mathdoc},
     volume = {302},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_302_a7/}
}
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N. P. Dolbilin. Delone sets in $\mathbb R^3$ with $2R$-regularity conditions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 176-201. http://geodesic.mathdoc.fr/item/TM_2018_302_a7/