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 one-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.