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 N X (T ) of radius T being finite for all T . We formulate conjectures relating slow growth of the patch-counting function N X (T ) to the set X having a non-trivial translation symmetry.A Delone set X 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. This is equivalent to the minimality of an associated topological dynamical system with R n -action. There is a lower bound for M X (T ) in terms of N X (T ), namely M X (T ) ≥ c(N X (T )) 1/n for some positive constant c depending on the Delone set constants r, R, but there is no general upper bound for M X (T ) purely in terms of N X (T ). The complexity of a repetitive Delone set X is measured by the growth rate of its repetitivity function M X (T ). For example, the function M X (T ) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if M X (T ) = O(T ) as T → ∞ and is densely repetitive if M X (T ) = O(N X (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 R n which has M X (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 quasicrystals'. § Current address: . ‡ The term 'local indistinguishability class' has also been proposed for this concept. † The suffix T is to indicate that the partial order derives from inclusion of translation atlases. In [26] this partial order is considered along with a partial order ≤I derived from inclusion of isometry atlases.

We prove that the set of visible points of any lattice of dimension n ≥ 2 has pure point diffraction spectrum, and we determine the diffraction spectrum explicitly. This settles previous speculation on the exact nature of the diffraction in this situation. Using similar methods we show the same result for the 1-dimensional set of kth-power-free integers with k ≥ 2. Of special interest is the fact that neither of these sets is a Delone set -each has holes of unbounded inradius. We provide a careful formulation of the mathematical ideas underlying the study of diffraction from infinite point sets.

The coincidence problem for planar patterns with N -fold symmetry is considered. For the N -fold symmetric module with N < 46, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case N ≥ 46 is briefly discussed and N = 46 is described explicitly.The results of the coincidence problem also solve the problem of colour lattices in two dimensions and its natural generalization to colour modules.

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Abstract. We consider the kth-power-free points in n-dimensional lattices and explicitly calculate their entropies and diffraction spectra. This is of particular interest since these sets have holes of unbounded inradius.