The Hyperbolic Metric on the Complement of the Integer Lattice Points in the Plane

A domain in the plane obtained by removing all integer lattice points admits the hyperbolic metric, which is the rank 2 Abelian cover of the once-punctured square tours. We compare the hyperbolic metric of this domain with a scaled Euclidean metric in the complement of the cusp neighborhoods. They are quasi-isometric. We investigate the best possible quasi-isometry constant relying on numerical experiment by Mathematica.