Locally Antipodal Delone Sets

The main subject of the talk is to expose recent results about so-called locally antipodal Delone sets in Euclidean space. Let $X$ be Delone set with parameters $r$ (the packing radius) and $R$ (the covering radius). As known one of the main goals of the local theory of regular point systems is to find in Delone set $X$ local conditions (rules) that guaranty the regularity / crystallographicity of the set. Delone set $X$ is called regular system if its symmetry group $G$ operates transitively ($X$ is $G$-orbit of a single point). Delone set is called crystal if $X$ is $G$-orbit of some finite set. A regular system is a very important particular case of a crystal. Remind here the most typical statements of the local theory (N.Dolbilin, M.Stogrin):


1) In plane: any Delone set in that all $4R$-custers (neighborhoods) are congruent is a regular system
2) In space of any dimension: the value $4R$ is unimprovable: for any $\varepsilon$ one can show a Delone set $X$ in which all $(4R-\varepsilon)$ clusters are the same but the $X$ is neither regular set nor crystallographic. 3) In $3D$ space: any Delone set with the same $10R$-clusters are regular sets. 4) In space of any dimension: there is an upper bound for the radius of identical clusters in a Delone set that guaranties the regularity of the set.

We call a Delone set $X$ locally antipodal if a $2R$-cluster at any point $x$ of $X$ is centrally symmetrical about the center $x$ of the cluster. In the talk, there will be discussed the following theorems which are true for any dimension.

Theorem 1. A locally antipodal Delone set is globally antipodal at any its point (see [2]).

Theorem 2. If two locally antipodal Delone sets $X$ and $Y$ have a $2R$-cluster in common then $X$ and $Y$ coincide totally (see [2]).

Theorem 3. A locally antipodal Delone set is the union of at most $2^d-1$ pairwise congruent and parallel lattices (see [2]).

Theorem 4. A locally antipodal Delone set with pairwise congruent $2R$-clusters is a regular system (see [1], [2]).

Theorems 1 and 4 can be used, in particular, for simplifying the $10R$-upperbound mentioned in p. 3). It is interesting to compare theorem 4 and assertion of p. 2) concerning the existence of irregular sets with the same $(4R-\varepsilon)$-clusters. The examples of non-regular sets with equivalent $(4R-\varepsilon)$-clusters that were found are not locally antipodal sets. This fact agrees well with theorem 4.

[1] N.P. Dolbilin, A criterion for crystal, and locally antipodal Delone sets, Proc. Int. Conf. “Quantum Topology”, Vestn. Chel. SU, 3(358) (2015), p.6-17.

[2] N.P. Dolbilin, A.N. Magazinov, Locally antipodal Delone sets, Russian Surv.,70:5(425) (2015), p.179–180.
This work is supported by the Russian Science Foundation under grant 14-11-00414.