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

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 However, a detailed proof remains unpublished for many years.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.