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A lattice in $\mathbb{R}^n$ is a discrete subgroup that spans $\mathbb{R}^n$. Recently I've been running into a similar sort of object consisting of more than $n$ vectors in $\mathbb{R}^n$ and their $\mathbb{Z}$-linear combinations.

I think this is best seen via an example.

Consider the boring lattice generated by $(1,0)$ and $(0,1)$. This is the good ol' standard coordinate lattice we know and love.

Now look at the 'non-discrete lattice' generated by $(1,0), (0,1), (1,\sqrt 2)$. This is not a discrete subgroup of $\mathbb{R}^2$ because it's 'horizontally dense,' i.e. if $(x,y)$ is in the 'lattice,' then there are infinitely many other points in $(x\pm \epsilon, y)$ in the 'lattice.' This is very different than the non-disrete lattice generated by $(1,0), (0,1), (\sqrt 2, \sqrt 2)$, which is everywhere dense.

These examples are a bit contrived, but in higher dimensions/more general fields, I have a hard time determining the resulting characteristics of the 'lattice.' For example, is it easy to tell how 'dense' the resulting 'lattice' is? Does this come up/are there known applications of this to, say, number theory?

To be specific:

  1. Is there a name for the concept of what I've been calling a 'non-discrete lattice?'
  2. Can you point me to reference material on 'non-discrete lattices?'
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  • $\begingroup$ What differentiates your definition of a non-discrete lattice from a finitely generated subgroup of $\mathbb{R}^n$? $\endgroup$ Commented Aug 13, 2013 at 10:58
  • $\begingroup$ @Alex Nothing. But lattices are also finitely generated subgroups of $\mathbb{R}^n$, and yet have a rich theory. $\endgroup$
    – davidlowryduda
    Commented Aug 13, 2013 at 11:51
  • $\begingroup$ Right, but to distinguish f.g. subgroups of $\mathbb{R}^n$ we made the distinction that their $\mathbb{Z}$-basis is linearly independent. What extra property do you want to impose on your "non-discrete lattices" to make it more restrictive than just f.g. subgroups. $\endgroup$ Commented Aug 13, 2013 at 11:53
  • $\begingroup$ It's not a direct relation and these are fuzzy memories, but I have vague recollection of there being some loose connections between these sorts of entities and Connes' noncommutative geometry (though that may be a false positive going through the Penrose lattices). Maybe someone with a better knowledge of exotic spaces can speak to that. $\endgroup$ Commented Aug 13, 2013 at 15:42

2 Answers 2

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One interesting thing this reminds me of is "equidistribution" issues, e.g., as Weyl, and going back to Kronecker. That is, in the simplest case, for irrational $\theta$, the collection of integer multiples of $\theta$ is equidistributed in the circle $\mathbb R/\mathbb Z$, in the sense that for (e.g., smooth, but $C^1$ more than suffices) periodic $f$, the limit of ${1\over N}\sum_{1\le n\le N} f(n\theta)$ is $\int_0^1 f$.

Similar results hold in higher dimension/rank, and in-between results exist, as illustrated in the question. That is, the analogous limit may be integration over a proper sub-torus.

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In the comments, Alex answered #1 as a "nondiscrete lattice" is really just a finitely generated group.

A similar type of thing does arise in number theory. For example, in the ring of integers $\mathbb{Z}[\sqrt{2}] \subset \mathbb{Q}[\sqrt{2}]$ is not a lattice inside of $\mathbb{R}$ as it is not discrete. A central philosophy of number theory is one must not forget Galois conjugates. Applying this in this case leads to the embedding $$(a+b\sqrt{2}) \rightarrow (a+b\sqrt{2}, a-b\sqrt{2}) \subset \mathbb{R} \times \mathbb{R}$$ and its image is a lattice. More generally this can be applied to any ring of integers (see here for the general technique and a classical application).

Also asked was the question:

"Does this come up/are there known applications of this to, say, number theory?"

I'm going to be bold and say the answer is "no", although I would love for someone who knows of an example to come forward. At the very least I can't think of any examples in the fields of number theory I'm most familiar with (classic algebraic number theory, modern algebraic number theory, Diophantine geometry). Admittedly I am less familiar with analytic number theory and (perhaps ironically) this is an area where I could see such subgroups most likely to be used.

Philosophically I think this is because number theory is a discrete subject almost by definition. More specifically, given a subgroup there are typically two things you can study: the subgroup itself and the quotient by the subgroup. When you want to study the subgroup itself, usually it's because it somehow relates to integers and thus should be discrete. Quotients are also used, especially in modern algebraic number theory with modular curves and their higher dimensional analogues. Again, here one needs the group to be discrete because there is a geometric structure which one wants to preserve on the quotient.

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