Post History
#2: Post edited
by
WheatWizard
·
2024-04-27T14:01:49Z (8 months ago)
Found another proof.
- 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.
A proof was given [here](https://math.stackexchange.com/questions/4906281/seeking-an-elementry-theorem-about-lattices#comment10472910_4906281), which seems 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*](https://archive.org/details/springer_10.1007-978-3-642-62035-5) and [*Sphere packings, lattices, and groups*](https://archive.org/details/spherepackingsla0000conw_b8u0), the former seems more relevant to me, but I wasn't able to find a statement of the theorem in either.
- 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](https://math.stackexchange.com/questions/4906281/seeking-an-elementry-theorem-about-lattices#comment10472910_4906281) and [here](https://www.ime.usp.br/~tausk/texts/DiscreteSubgroups.pdf), 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*](https://archive.org/details/springer_10.1007-978-3-642-62035-5) and [*Sphere packings, lattices, and groups*](https://archive.org/details/spherepackingsla0000conw_b8u0), the former seems more relevant to me, but I wasn't able to find a statement of the theorem in either.
#1: Initial revision
by
WheatWizard
·
2024-04-27T13:29:32Z (8 months ago)
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.
A proof was given [here](https://math.stackexchange.com/questions/4906281/seeking-an-elementry-theorem-about-lattices#comment10472910_4906281), which seems 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*](https://archive.org/details/springer_10.1007-978-3-642-62035-5) and [*Sphere packings, lattices, and groups*](https://archive.org/details/spherepackingsla0000conw_b8u0), the former seems more relevant to me, but I wasn't able to find a statement of the theorem in either.