Diffraction from visible lattice points and k-th power free integers
Abstract
We prove that the set of visible points of any lattice of dimension at least 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, see math-ph/9903046 and references therein. Using similar methods we show the same result for the 1-dimensional set of k-th power free integers with k at least 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.
Cite
@article{arxiv.math/9906132,
title = {Diffraction from visible lattice points and k-th power free integers},
author = {Michael Baake and Robert V. Moody and Peter A. B. Pleasants},
journal= {arXiv preprint arXiv:math/9906132},
year = {2007}
}
Comments
45 pages, with minor corrections and improvements; dedicated to Ludwig Danzer on the occasion of his 70th birthday