Simultaneous visibility in the integer lattice

Two lattice points are visible from one another if there is no lattice point on the open line segment joining them. Let $S$ be a finite subset of $\mathbb{Z}^k$. The asymptotic density of the set of lattice points, visible from all points of $S$, was studied by several authors. Our main result is an improved upper bound on the error term. We also find the Schnirelmann density of the set of visible points from some sets S. Finally, we discuss these questions from the point of view of ergodic theory.