Delone sets in ℝ3: Regularity Conditions

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 However, a detailed proof remains unpublished for many years.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

This is a preview of subscription content, log in to check access.

References

  1. 1.

    B. N. Delone, N. P. Dolbilin, M. I. Shtogrin, and R. V. Galiulin, “A local criterion for regularity of a system of points,” Sov. Math. Dokl., 17, 319–322 (1976).

    MATH  Google Scholar 

  2. 2.

    N. P. Dolbilin, “Criterion for a crystal and locally antipodal Delone sets,” Vestn. Chelyabinsk. Gos. Univ., No. 3 (358), 6–17 (2015).

  3. 3.

    N. Dolbilin, “Delone sets with congruent clusters,” Struct. Chem., 26, No. 6 (Dedicated to the 75th anniversary of Academician V. Ya. Schevchenko), 1725–1732 (2016).

  4. 4.

    N. Dolbilin, “Delone sets: Local identity and global order,” in: Discrete Geometry and Symmetry. Dedicated to Károly Bezdek and Egon Schulte on the Occasion of Their 60th Birthdays Springer, Berlin (2017), pp. 109–125; arXiv:1608.06842.

  5. 5.

    N. P. Dolbilin and A. N. Magazinov, “Locally antipodal Delaunay Sets,” Russ. Math. Surv., 70, No. 5, 958–960 (2015).

    MathSciNet  Article  Google Scholar 

  6. 6.

    N. P. Dolbilin and A. N. Magazinov, “Uniqueness theorem for locally antipodal Delaunay sets,” Proc. teklov Inst. Math., 294, 215–221 (2016).

    MathSciNet  Article  Google Scholar 

  7. 7.

    R. Feynman, R. Leighton, and M. Sands, Feynman Lectures on Physics, Vol. II, Addison-Wesley (1964).

  8. 8.

    M. Shtogrin, “On limiting the order of the axis of a spider in a locally regular Delone system,” in: Geometry, Topology, Algebra and Number Theory, Applications. The International Conference dedicated to the 120th anniversary of Boris Nikolaevich Delone (1890–1980) (Moscow, August 16–20, 2010), Abstracts [in Russian], Steklov Math. Inst., Moscow (2010), pp. 168–169

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to N. P. Dolbilin.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 6, pp. 115–141, 2016.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dolbilin, N.P. Delone sets in ℝ3: Regularity Conditions. J Math Sci (2020). https://doi.org/10.1007/s10958-020-04909-8

Download citation