  This paper considers the problem of characterizing the simplest discrete
point sets that are aperiodic, using invariants based on topological dynamics.
A Delone set whose patch-counting function N(T), for radius T, is finite for
all T is called repetitive if there is a function M(T) such that every ball of
radius M(T)+T contains a copy of each kind of patch of radius T that occurs in
the set. This is equivalent to the minimality of an associated topological
dynamical system with R^n-action. There is a lower bound for M(T) in terms of
N(T), namely N(T) = O(M(T)^n), but no general upper bound.
  The complexity of a repetitive Delone set can be measured by the growth rate
of its repetitivity function M(T). For example, M(T) is bounded if and only if
the set is a crystal. A set is called is linearly repetitive if M(T) = O(T) and
densely repetitive if M(T) = O(N(T))^{1/n}). 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 reverse direction, we construct a
repetitive Delone set in R^n which has
  M(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".
