# Seeking a theorem about lattices

I am looking for a reference on the following theorem, or an equivalent statement:

Let $\Lambda$ be an embedding of a free $\mathbb{Z}$-module in $\mathbb{R}^d$. If the rank of $\Lambda$ is greater than $d$ then $\Lambda$ is not discrete.

I have proofs here and here, which both seem correct to me. However I'm ideally looking for a published source for the theorem.

I skimmed through the early chapters of *An introduction to the geometry of numbers* and *Sphere packings, lattices, and groups*, the former seems more relevant to me, but I wasn't able to find a statement of the theorem in either.

## 1 answer

Neukirch - Algebraic Number Theory proposition 4.2 states that a subgroup of $\mathbb{R}^n$ is discrete if and only if it is a lattice. A lattice has a linearly independent basis by definition, so it has rank at most $n$.

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