# Questions tagged [euclidean-lattices]

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### Random walk to visible lattice points

Consider a random walk from the $\mathbb{Z}^2$ origin $(0,0)$ to visible (not blocked) lattice points $p$, with a parameter $r$ a given radius of a circle centered on $p$. With $p$ the previous point, ...
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### Illumination from visible lattice points with inverse square intensity

It is well known that the number of $\mathbb{Z}^2$ lattice points visible from the origin is $6/\pi^2$, about $61$%. See, e.g., What fraction of the integer lattice can be seen from the origin?. I am ...
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### Does $2$ variable linear Diophantine equation in $NC$ imply $2$ dimensional shortest vector is in $NC$?

Consider the Linear Diophantine in known $a,b,c\in\mathbb Z$ $$ax+by=c.$$ Above can be solve by Extended Euclidean which is not in $NC$ as far as we know. It is clear if Extended Euclidean is in $NC$ ...
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1 vote
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### Existence of some lattice path connecting all given lattice paths

My daily work concerns analysis on metric spaces and some time ago it turned out that the problem I am dealing with boils down to a certain combinatorial problem. I've checked a lot of examples and it ...
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### Counting symmetric convex bodies with no nonzero lattice point in the interior

In order to estimate the size of the torsion in the algebraic $K$-groups of $\mathbb Q$ one needs to understand the homology of $\mathrm{GL}_n(\mathbb Z)$, or alternatively, the homology of the space ...
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### The lattice handshake number ("nearly kissing" number)?

Update: I'm happy to say that this question has been made essentially obsolete by the breakthrough result of Serge Vlăduţ, who showed that the kissing number is exponentially large: https://arxiv.org/...
1 vote
145 views

### Densest sphere packing for a given lattice

There is a great deal of fascinating work on the sphere packing problem. I was wondering if there exist methods to find the densest possible packing of a sublattice of a given lattice. In particular, ...
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### Bound on the fundamental region of a sublattice of a Niemeier Lattice

I am working with an 18 dimensional lattice, $W$ say, primitively embedded into a given Niemeier lattice $\mathcal{N}_i$. I am trying to figure out the following: what is the smallest possible ...
Consider a positive-definite $2n$-dimensional lattice with minimum norm $\mu$. It is sometimes possible to find a generating set of minimal vectors for the lattice such that the Gram matrix takes the ...
The possible configurations of minimal vectors for a 4-dimensional lattice are known for ages, but what about symplectic lattices ? If a 4-dimensional symplectic lattice $\Lambda$ has two minimal ...