Simultaneous visibility in the integer lattice

Let S be a finite subset of the integer lattice Z^d with d>1. A lattice point x is said to be visible from the set S if, for every s in S, there is no other lattice point on the line segment joining x and s. The density of the set of points visible from S has been known since 1960, and error terms have been studied over time under conditions on S. We say that a positive integer L is exceptional if the proportion of visible points in the box [1, L]^k is strictly below the density.

We will present an improved upper bound for the error term, along with results on exceptional points, Schnirelmann density, and related conjectures and open questions about visible points. If time permits, we will discuss ergodic properties of the visible lattice points.

Joint work with Daniel Berend and Andrew Pollington.