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Multiregular point systems. (English) Zbl 0915.51017
Summary: This paper gives several conditions in geometric crystallography which force a structure \(X\) in \(\mathbb{R}^n\) to be an ideal crystal. An ideal crystal in \(\mathbb{R}^n\) is a finite union of translates of a full-dimensional lattice. An \((r,R)\)-set is a discrete set \(X\) in \(\mathbb{R}^n\) such that each open ball of radius \(r\) contains at most one point of \(X\) and each closed ball of radius \(R\) contains at least one point of \(X\). A multiregular point system \(X\) is an \((r,R)\)-set whose points are partitioned into finitely many orbits under the symmetry group Sym\((X)\) of isometries of \(\mathbb{R}^n\) that leave \(X\) invariant. Every multiregular point system is an ideal crystal and vice versa.
We present two different types of geometric conditions on a set \(X\) that imply that it is a multiregular point system. The first is that if \(X\) “looks the same” when viewed from \(n+2\) points \(\{y_i:1\leq i\leq n+2\}\), such that one of these points is in the interior of the convex hull of all the others, then \(X\) is a multiregular point system. The second is a “local rules” condition, which asserts that if \(X\) is an \((r,R)\)-set and all neighborhoods of \(X\) within distance \(\rho\) of each \({\mathbf x}\in X\) are isometric to one of \(k\) given point configurations, and \(\rho\) exceeds \(CRk\) for \(C=2(n^2+1) \log_2(2R/r +2)\), then \(X\) is a multiregular point system that has at most \(k\) orbits under the action of \(\text{Sym}(X)\) on \(\mathbb{R}^n\). In particular, ideal crystals have perfect local rules under isometries.

51P05 Classical or axiomatic geometry and physics (should also be assigned at least one other classification number from Sections 70-XX–86-XX)
82D25 Statistical mechanical studies of crystals
51F15 Reflection groups, reflection geometries
51M05 Euclidean geometries (general) and generalizations
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