Families of lattices with an unbounded number of unit vectors

TL;DR

This paper introduces three families of 4-dimensional lattices, each with a finite number of unit vectors, but collectively exhibiting an unbounded number of unit vectors across the families, highlighting novel lattice structures.

Contribution

The paper constructs three specific lattice families in four dimensions, demonstrating unbounded unit vectors despite each having finite unit vectors individually.

Findings

Each lattice family has a finite number of unit vectors.

The total number of unit vectors across the three families is unbounded.

Lattice $L_3$ is identified as the Moser lattice.

Abstract

3 families of 4-dimensional lattices $L_k, M_k, M_k / 2 \subset \mathbb{R}^2$ are defined. Each lattice is defined by 2 quadratic extensions and has a \emph{finite} number of unit vectors, but the number of unit vectors in each of the 3 familes is \emph{unbounded}. $L_3$ is the Moser lattice.