On the diffraction spectrum of the set of visible points in lattices and certain cut-and-project sets

Let $k\geq 2$ be a positive integer.

It is known that the set of visible lattice points from the origin in $\mathbb{Z}^k$ has a translation bounded pure point diffraction spectrum.

We investigate these properties for sets of points simultaneously visible from a finite set of lattice points $ \{\mathbf{x}_1,\dots,\mathbf{x}_n\} \subseteq \mathbb{Z}^k$.

We provide explicit formulas for the coefficients of the diffraction spectrum.

Additionally, we generalize our procedure to show that the set of visible points from the origin in certain classes of cut-and-project sets has a translation bounded pure point diffraction spectrum.