Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as $mathscr B$-free numbers w
ith Sarnaks conjecture on the `randomness of the Mobius function, another the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.
We revisit the visible points of a lattice in Euclidean $n$-space together with their generalisations, the $k$th-power-free points of a lattice, and study the corresponding dynamical system that arises via the closure of the lattice translation orbit
. Our analysis extends previous results obtained by Sarnak and by Cellarosi and Sinai for the special case of square-free integers and sheds new light on previous joint work with Peter Pleasants.
We consider the problem of distinguishing convex subsets of $n$-cyclotomic model sets $varLambda$ by (discrete parallel) X-rays in prescribed $varLambda$-directions. In this context, a `magic number $m_{varLambda}$ has the property that any two conve
x subsets of $varLambda$ can be distinguished by their X-rays in any set of $m_{varLambda}$ prescribed $varLambda$-directions. Recent calculations suggest that (with one exception in the case $n=4$) the least possible magic number for $n$-cyclotomic model sets might just be $N+1$, where $N=operatorname{lcm}(n,2)$.
