Simultaneous visibility in the integer lattice

TL;DR

This paper investigates the density and visibility properties of lattice points in integer grids, providing improved bounds, calculating densities for specific sets, and exploring ergodic theory perspectives.

Contribution

It offers an improved upper bound on the error term for the asymptotic density of points visible from a set in the integer lattice and analyzes their densities and ergodic properties.

Findings

Improved upper bound on the error term for lattice point visibility density.

Calculated Schnirelmann density for certain sets of visible points.

Discussed ergodic theory implications of visibility in integer lattices.

Abstract

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.