Bounds for solid angles of lattices of rank three

TL;DR

This paper establishes sharp bounds on the solid angles spanned by bases of rank 3 lattices, with implications for lattice geometry and the kissing number problem.

Contribution

It provides the first sharp absolute bounds for solid angles in rank 3 lattices, extending to a broader class than well-rounded lattices, and relates these bounds to lattice minimal vectors.

Findings

Sharp bounds $C_1$ and $C_2$ for solid angles in rank 3 lattices.

The minimal solid angle configuration is achieved by the face centered cubic lattice.

The results connect lattice geometry with spherical triangle areas and the kissing number problem.

Abstract

We find sharp absolute constants $C_1$ and $C_2$ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval $[C_1,C_2]$. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in $\mathbb R^N$ with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in $\mathbb R^3$. Such spherical configurations come up in connection with the kissing number problem.