# Questions tagged [lattices]

Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])

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### Lattice points in hypercubes

Let $(\Lambda_n)$ be a family of lattices, $\Lambda_n \subset \mathbb{Z}^n$, with $\det\Lambda_n \sim n$ as $n \to \infty$ (meaning $\lim_{n\to\infty} n^{-1} \det\Lambda_n = 1$). I am ...
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### Finding a superbase in a lattice of Voronoi first kind

An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that $\{b_1,\ldots,b_n \}$ is a ...
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### Number of points in a lattice and an oblong box

I have a very simple question in geometry of numbers. (It is a slight modification of Counting points on the intersection of a box and a lattice .) There's a bound I can easily prove, and it's good ...
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### Lattices with no roots and spread out shells

I am looking for lattices with the following properties: The lattice has no roots. The norm (squared length) of the second shortest vectors should be at least twice as large as the norm of the ...
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### Edges of the contact polytope of the Leech lattice

Let $P\subset\Bbb R^{24}$ be the contact polytope of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda_{24}$. Question: What are the edges of $P$? Let'...
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### Representation of the space of lattices in $\Bbb R^n$

The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
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### Submersion to $T^{2}$

Let $M$ be a $2n$-dimensional compact and connected manifold. Suppose there is $\Omega\in\Omega^{1}(M,\mathbb{C})$ a closed complex form whose real and imaginary parts represent linearly ...
### Selfsimilar lattices in $\mathbb R^d$
Let $\Lambda\subset \mathbb R^d$ a discrete subgroup, up to diminishing $d$ we assume it is of the form $A\mathbb Z^d$ with $A\in GL(d)$. Up to dilation we assume that the shortest vector in \$\Lambda\...