The Local Theory for Regular Systems in the Context of t-Bonded Sets

Abstract

Abstract:The main goal of the local theory for crystals developed in the last quarter of the 20th Century by a geometry group of Delone (Delaunay) at the Steklov Mathematical Institute is to find and prove the correct statements rigorously explaining why the crystalline structure follows from the pair-wise identity of local arrangements around each atom. Originally, the local theory for regular and multiregular systems was developed with the assumption that all point sets under consideration are (r, R)-systems or, in other words, Delone sets of type (r, R) in d-dimensional Euclidean space. In this paper, we will review the recent results of the local theory for a wider class of point sets compared with the Delone sets. We call them t-bonded sets. This theory, in particular, might provide new insight into the case for which the atomic structure of matter is a Delone set of a "microporous" character, i.e., a set that contains relatively large cavities free from points of the set.

Bounds for the Regularity Radius of Delone Sets

“…In Sect. 4 we extend our study of the regularity radius to the family of t-reachable sets, see also [9,13]. In Theorems 4.1 and 4.2 we prove an upper bound O(d 2 log 2 d)R for the family of t-reachable Delone sets provided t ≤ ar for some fixed constant a.…”

“…Our discussion of t-reachable points is inspired by the concept of t-bonded sets introduced in Dolbilin [13] and developed in Bouniaev & Dolbilin [9]. A Delone set X is called t-bonded for a parameter t > 0 if every two points x, y ∈ X can be connected by a finite chain x = x 0 , .…”

On the Regularity Radius of Delone Sets in $${\mathbb {R}}^3$$

“…Our discussion of t-reachable points is inspired by the concept of t-bonded sets introduced in Dolbilin [8] and developed in Bouniaev & Dolbilin [4]. A Delone set X is called t-bonded for a parameter t > 0 if every two points x, y ∈ X can be connected by a finite chain…”

“…In Section 4 we extend our study of the regularity radius to the family of t-reachable sets, see also [4,8]. In Theorems 4.1 and 4.2 we prove an upper bound O(d 2 log d)R for the family of t-reachable Delone sets provided t ≤ ar for some fixed constant a.…”