Lattice of Gauss and Eisenstein Integers
- Z is a 1D lattice
- Gaussian and Eisenstein integers are 2D lattices
- But the golden integers (for example) are dense on the real line.
Are there rings of integers which have 3D, 4D, ... lattices?
Here is a plot of $(a + \tfrac{1}{2}(1+\sqrt{5})b,a + \tfrac{1}{2}(1-\sqrt{5})b)$ for $-10\le a,b \le 10$.
which is the lattice corresponding to the golden integers, if I understand correctly. The green points represent rational integers and the blue points represent multiples of $\varphi$.
The example of the golden integers (and more generally rings of integers in quadratic number fields with positive discriminant) shows that the single embedding into $\mathbb{R}$ is inadequate. Instead, if $K = \mathbb{Q}(\sqrt{d}), d > 0$ then the appropriate way to embed $K$ as a lattice in the plane is to look at both embeddings $\sigma_1, \sigma_2 : K \to \mathbb{R}$. The first one sends $\sqrt{d}$ to $\sqrt{d}$ and the second one sends $\sqrt{d}$ to $-\sqrt{d}$. Together they give an embedding $(\sigma_1, \sigma_2)$ of $K$ into $\mathbb{R}^2$, and relative to this embedding the ring of integers $\mathcal{O}_K$ really is a lattice.
More generally, if $K$ is a number field of degree $n$, then there are $n = r + 2s$ embeddings $\sigma_i : K \to \mathbb{C}$, $r$ of which have image in $\mathbb{R}$ and $2s$ of which have image outside of $\mathbb{R}$, which come in complex conjugate pairs. Here the appropriate generalization of the above embedding is to use all of the real embeddings $\sigma_1, ... \sigma_r$ and one representative of each complex conjugate pair of complex embeddings $\sigma_{r+1}, ... \sigma_{r+s}$. This gives an embedding $K \to \mathbb{R}^r \times \mathbb{C}^s$, and embedding $\mathbb{C}$ into $\mathbb{R}^2$ gives an embedding $K \to \mathbb{R}^n$.
Relative to this embedding, it's a standard exercise that $\mathcal{O}_K$ is a lattice in $\mathbb{R}^n$ of rank $n$. This is the standard construction used to prove the finiteness of the class group and Dirichlet's unit theorem, and details can be found in any book on algebraic number theory.