Repetitive Delone Sets and Quasicrystals

This paper studies the problem of characterizing the simplest aperiodic discrete point sets, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X −X is locally finite. Such sets are characterized by their patch-counting function NX(T ) of radius T being finite for all T . We formulate conjectures relating slow growth of the patch-counting function NX(T ) to the set X having a nontrivial translation symmetry. A Delone set X of finite type is repetitive if there is a function MX(T ) such that every closed ball of radius MX(T ) + T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with Rn-action. There is a lower bound for MX(T ) in terms of NX(T ), namely MX(T ) ≥ c(NX(T ))1/n for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for MX(T ) purely in terms of NX(T ). The complexity of a repetitive Delone set X is measured by the growth rate of its repetitivity function MX(T ). For example, the function MX(T ) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if MX(T ) = O(T ) as T → ∞ and is densely repetitive if MX(T ) = O(NX(T )) 1/n as T → ∞. We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, i.e. 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. In the reverse direction, we construct a repetitive Delone set X in Rn which has MX(T ) = O(T (log T ) 2/n(log log log T )4/n), but does not have uniform patch frequencies. Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets, and we propose considering them as a notion of “perfectly ordered quasicrystal”.