The mathematical conditions for the origin of long‐range order or crystallinity in ideal crystals are one of the very fundamental problems of modern crystallography. It is widely believed that the (global) regularity of crystals is a consequence of `local order', in particular the repetition of local fragments, but the exact mathematical theory of this phenomenon is poorly known. In particular, most mathematical models for quasicrystals, for example Penrose tiling, have repetitive local fragments, but are not (globally) regular. The universal abstract models of any atomic arrangements are Delone sets, which are uniformly distributed discrete point sets in Euclidean d space. An ideal crystal is a regular or multi‐regular system, that is, a Delone set, which is the orbit of a single point or finitely many points under a crystallographic group of isometries. The local theory of regular or multi‐regular systems aims at finding sufficient local conditions for a Delone set X to be a regular or multi‐regular system. One of the main goals is to estimate the regularity radius for Delone sets X in terms of the radius R of the largest `empty ball' for X. The celebrated `local criterion for regular systems' provides an upper bound for for any d. Better upper bounds are known for d ≤ 3. The present article establishes the lower bound for all d, which is linear in d. The best previously known lower bound had been for d ≥ 2. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent (2dR − ϵ)‐clusters, which are not regular systems. The two‐ and three‐dimensional constructions are illustrated by examples. In addition to its fundamental importance, the obtained result is also relevant for the understanding of geometrical conditions of the formation of ordered and disordered arrangements in polytypic materials.

Abstract:The concept of t-bonded sets was briefly introduced by the second author in 1976 under the name of dconnected sets, though it has not received due consideration. This concept is a generalization of the concept of Delone (r, R)-systems. In light of the developments in the local theory for crystals that occurred since 1976 and demands in chemistry and crystallography, we believe the local theory for t-bonded sets deserves to be developed to describe materials whose atomic structures is multi-regular "microporous" point set. For a better description of such "microporous" structures it is worthwhile to take into consideration a parameter that represents atomic bonds within the matter. The overarching goal of this paper is to prove that analogous local conditions that guarantee that a Delone set is a regular (or multi-regular) system also guarantee that a t-bonded set is a regular (or multi-regular) t-bonded system.

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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.

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