Density and holes of the visible points in certain cut-and-project sets

We study the set of visible points from the origin in certain cut-and-project sets, specifically the density of the set. We generalize to all real quadratic number fields the formulas for the density found by G. Hammarhjelm, and the corresponding error bounds proved by I. Gringlaz, R. Kumar, B. Weiss. We also find the possible densities of the set of visible points in the cut-and-project set, with a constant generic lattice, and a changing window with constant volume. We also study bounds for the distance of large holes in the set of visible points of certain cut-and-project sets, generalizing theorems originally proved by P. Erdős for Z².