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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