On lattice coverings of Nil space by congruent geodesic balls

TL;DR

This paper investigates lattice coverings in Nil space using geodesic balls, providing density estimates, a conjecture for optimal arrangements, and a specific covering density value.

Contribution

It introduces the concept of covering density in Nil space, estimates bounds, and proposes a conjecture for the optimal ball arrangement with a specific density.

Findings

Estimated covering density: approximately 1.42900615

Formulated a conjecture for the least dense lattice-like covering

Provided upper and lower bounds for covering density

Abstract

The Nil geometry, which is one of the eight 3-dimensional Thurston geometries, can be derived from {W. Heisenberg}'s famous real matrix group. The aim of this paper to study {\it lattice coverings} in Nil space. We introduce the notion of the density of considered coverings and give upper and lower estimations to it, moreover we formulate a conjecture for the ball arrangement of the least dense lattice-like geodesic ball covering and give its covering density $\Delta\approx 1.42900615$. The homogeneous 3-spaces have a unified interpretation in the projective 3-sphere and in our work we will use this projective model of the Nil geometry.