Families of lattices with an unbounded number of unit vectors
Helmut Ruhland
TL;DR
The paper addresses the existence of finitely generated subgroups of $\mathbb{R}^2$ with infinitely many unit vectors by constructing three 4-dimensional lattice families $L_k$, $M_k$, and $M_k/2$ via two quadratic extensions, where each family has a finite unit-vector set but the total shell size is unbounded across the family. It develops explicit lattice constructions using hexagonal and square sublattices, derives unit-vector counts from representations in quadratic forms (notably $H(a,b)=a^2+ab+b^2$ and $S(a,b)=a^2+b^2$), and connects growth to prime decompositions in related rings (Eisenstein and Gaussian integers). The work also analyzes symmetry groups, showing $D_{12}$ or $D_{8}$ in general, with exceptional cases $L_{1/2}$ and $M_1$ attaining $D_{24}$ due to self-conjugacy and quartic extensions, and clarifies the algebraic structure via Galois theory and root-of-unity representations. Overall, it demonstrates unbounded shell sizes within 4D lattice families and elucidates the symmetry structures that govern these unbounded families.
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.