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Repetitive Delone sets and quasicrystals. (English) Zbl 1062.52021
A Delone set, or \((r,R)\)-set, is a discrete set \(X\subset \mathbb{R}^n\) such that each open ball of radius \(r\) in \(\mathbb{R}^n\) contains at most one point of \(X\) and each closed ball of radius \(R\) in \(\mathbb{R}^n\) contains at least one point of \(X\). A Delone set of finite type is a Delone set \(X\) such that \(X-X\) is locally finite, that is, \(X-X\) is a set with only finitely many points in any bounded region. Such sets are characterized by their patch-counting function \(N_X(T)\) of radius \(T\) being finite for all \(T\). A Delone set of finite type is repetitive if there is a function \(M_X(T)\) such that every closed ball of radius \(M_X(T)+T\) contains a complete copy of each kind of patch of radius \(T\) that occurs in \(X\).
The complexity of a repetitive Delone set \(X\) is measured by the growth rate of its repetitivity function \(M_X(T)\). A set \(X\subset \mathbb{R}^n\) is a periodic crystal if and only if the function \(M_X(T)\) is bounded. A set \(X\subset \mathbb{R}^n\) is linearly repetitive if \(M_X(T)=O(T)\) as \(T\to \infty \) and it is densely repetitive if \(M_X(T)=O(N_X(T))^{1/n}\) as \(T\to \infty \).
The authors show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, that is, the associated topological dynamical system is strictly ergodic. It follows that such sets are diffractive, in the sense of having a well-defined diffraction measure.
A set with no translation symmetries is called an aperiodic set. The general problem motivating this paper is that of characterizing the ‘simplest’ locally finite point sets \(X\subset \mathbb{R}^n\) that are aperiodic. The authors study this problem for Delone sets and the proposed notions of simplicity involve constraints on the patches in the set and invariants from topological dynamics. The class of Delone sets of finite type give useful models for quasicrystalline materials (materials whose x-ray diffraction spectra have sharp spots, indicating some long-range order in the atomic structure, but which lack a lattice of periods).

MSC:
52C23 Quasicrystals and aperiodic tilings in discrete geometry
37A25 Ergodicity, mixing, rates of mixing
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
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