Diffraction from visible lattice points and kth power free integers

Abstract

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.

Dynamical Properties of k-Free Lattice Points

“…For the cases k = 1 and d ≥ 2 as well as k ≥ 2 and d = 1, the construction is given in [1]; see also [20], Ch. 5a for an alternative description.…”

What is a crystal?

“…Baake and several co-workers [32][33][34][35][36] are currently performing a systematic study whose purpose is to charac-terize which distributions of matter diffract to produce a pure point component in their spectrum, and thus can qualify as possessing long-range order. In some cases it is even difficult to determine whether the Fourier transform of a structure exists, in the sense that it has a unique infinite volume limit.…”

Untitled

“…One way to establish the existence of the limit is through the pointwise ergodic theorem, compare [15], if such methods apply. If not, explicit convergence proofs will be needed, as is apparent from known examples [5] and counterexamples [40].…”

“…This is certainly the case if one can refer to the ergodicity of the underlying distribution of scatterers [15,26,53], in particular, if their positional arrangement is linearly repetitive [40]. However, this is often difficult to assess in situations without underlying ergodicity properties, see [5] for an example. On the other hand, even if the image is uniquely defined, one still wants to know whether it contains Bragg peaks or not, or if there is any diffuse scattering present in it.…”