12
votes
Accepted
what is the number of paths returning to 0 on the hexagonal lattice
This is answered by Ian Agol here, with the reference "All Roads Lead to Rome-Even in the Honeycomb World", Brani Vidakovic, Amer. Statist. 48 (1994) no. 3, 234-236.
An exact formula is
$$ p(n) = \...
- 2,175
9
votes
Accepted
Joining the $2^k$ points of $\{0,1\}^k$ with the shortest tree
I suspect that the optimum, for a cube of side length $2$, is $2^k \sqrt{3} - 2 \sqrt{3}+2$. Note that the optimum if we use edges of the cube is $2 (2^k-1)$, so this is better by a factor of roughly $...
- 156k
8
votes
Accepted
Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$
Let $L$ denote the root lattice of $E_8$. The group $W(E_8)$ acts (linearly) on $L/2L \cong \mathbb{F}_2^8$, and hence as a permutation of $2^8=256$ elements. This permutation action has the following ...
- 54.6k
7
votes
Tri-coloring of E8 lattice? Why is the Gram matrix of E8 not unique?
The Gram matrix depends on the choice of a lattice basis (it's just the matrix of inner products of the basis vectors), which means all lattices in two or more dimensions have infinitely many Gram ...
- 16.8k
7
votes
what is the number of paths returning to 0 on the hexagonal lattice
If you want a rough answer, it is something of the order of $\frac{3^n}{n}$. This random paths are easier than self avoiding walks, you can think of these paths in this way: If you consider the even ...
- 1,101
7
votes
Accepted
Illumination from visible lattice points with inverse square intensity
Let's look at the number of the representations of $n$ as $a^2 + b^2$ with coprime $a, b$, and denote it $v(n)$. Our sum is then $\sum_{n \leq r^2} {\frac{v(n)}{n}}$.
We can see that $\sum_{d^2 | n} v(...
- 3,319
6
votes
Accepted
Number of planes generated by integer vectors
For $k=d-1$ this is a result of Bárány-Harcos-Pach-Tardos (2001). See Theorem 3 in the preprint version or the published version.
- 105k
5
votes
Accepted
Can we count the number of integer lattice points in this case?
An exact formula can be given for what you seek in terms of the functions giving point counts for balls. However, those functions are only known to approximate accuracy. The accuracy is good, but ...
- 30.1k
5
votes
Accepted
The lattice handshake number ("nearly kissing" number)?
I'm happy to say that this question has been answered by the breakthrough result of Serge Vlăduţ, who showed that the kissing number is exponentially large: https://arxiv.org/abs/1802.00886 !
I.e., ...
- 1,016
5
votes
Accepted
On shortest vector problem
Yes you can. I'll write $\lambda_1(L) = \min_{l\in L\setminus \{0\}} \lVert l\rVert_2$.
I will also identify a lattice with any of its basis, e.g. I will write $\lambda_1(B)$ where $B$ is a lattice ...
- 2,183
4
votes
Lattice with Voronoi cell inside a circle
Your conjecture is correct. The condition on $\Lambda^*$ means that
the covering radius of $\Lambda^*$ is at most $1$, and it is known that
the best covering lattice is hexagonal, which makes $\...
- 79.9k
4
votes
Can we count the number of integer lattice points in this case?
Given any subset $S$ of $\{1,\dots,d\}$, let $L_S(r)$ be the set of lattice points within distance $r$ of the origin with the property that their $j$th coordinates, for all $j\notin S$, equal $0$. For ...
- 12.8k
4
votes
Accepted
Proof of generalized Siegel's mean value formula in geometry of numbers
Such a generalization (roughly) exists, known as Rodger's Integration Formula.
See Section 1.2 of Seungki Kim's Dissertation for a reference.
Theorems 1.2 and 1.3 are of interest.
Theorem 1.2: (...
- 2,183
4
votes
I believe that all facets of a Voronoi-cell of a lattice are centerally symmetric. Is my argument correct? Is this true?
This is a community wiki answer based on the comments by alesia and Richard Stanley.
You already gave the hardest part of the argument. To finish it off, note that translating by $\mathbf{l}$ the ...
3
votes
The range of each of successive minima for all unimodular lattices
While I do not have an answer, there are many pointers one can give that might be useful, too many for comments.
First, one can say better than $B_1 < \infty$.
It is the Hermite constant, and one ...
- 2,183
3
votes
Accepted
"Sparse" Theta Series
Won't the following argument show that the difference between successive exponents can never be bounded away from zero no matter how clever you try to be in selecting $(a,b)$?
The idea is to consider ...
- 1,362
3
votes
What fraction of the integer lattice can be seen from the origin?
Here is an elementary argument using Euler products. I'm not sure if the initial probability part is rigorous, but it's a heuristic argument at least. The probability of a prime $p$ dividing a ...
- 2,075
3
votes
Accepted
On necessary condition for no integer points in polytope
The answer is $no$.
In the plane, the circle of diameter $d$ greater than $1$ but smaller than $\sqrt2$ and centered at $({1\over2},{1\over2})$ contains no point with integer coordinates, and the ...
- 7,276
3
votes
What is the spinor genus of the Leech lattice?
The genus of every even unimodular $\mathbb Z$-lattice has only one spinor genus. See O'Meara, particularly Example 102:10.
- 646
3
votes
Accepted
Existence of some lattice path connecting all given lattice paths
Let us draw a checkered table with $N$ columns and $n$ rows, such that the cell $(i,j)$ (that is, the one in the $i$th row and the $j$th column) contains the number $q_i^j$. Paint all cells with ...
- 23.7k
3
votes
What's the name of this constant similar to that of Hermite's?
It is known that
$$
(\sqrt{n}/12)^n\leq \delta_n \leq n^n.
$$
See Theorem 1.41 and Lemma 1.42 of Rothvoss' notes.
- 2,183
2
votes
Accepted
What is the spinor genus of the Leech lattice?
see Mass of spinor genus, positive integral quadratic forms for the statement that spinor genera in the same genus have the same mass, answer by Schulze-Pillot.
The spinor genus count being a ...
- 25.7k
2
votes
Densest sphere packing for a given lattice
Unfortunately, this is equivalent to the question of finding the densest lattice sphere packing in general. In particular, for any lattices $L$ and $L'$, there exists a sequence of sublattices $L_n$ ...
- 1,016
2
votes
Accepted
Integer points spanned by real, rational and integer combination of integer vectors
We always have $\mathcal L_\Bbb Q=\mathcal L_\Bbb R:$ For any particular $v \in\Bbb Z^n$ consider how we determine $\{u \in \Bbb R^n \mid uB=v\}.$ The method, if the set is nonempty, will yield one or ...
- 30.1k
2
votes
Integer points spanned by real, rational and integer combination of integer vectors
This is only a partial answer.
$\mathcal{L}_\mathbb{Q}=\mathcal{L}_\mathbb{R}$ for all $B$. Decompose $\mathbb{R}$ as a $\mathbb{Q}$-vector space into $\mathbb{Q}\oplus V$. Any $u\in\mathbb{R}^k$ can ...
- 1,593
2
votes
Accepted
Distance for $GL_n(\mathbb{R})/GL_n(\mathbb{Z})$
Fixing any right invariant metric $D$ on $G=\text{GL}_n(\mathbb{R})$ you get a metric $d$ on $G/\Gamma$ ($\Gamma=\text{GL}_n(\mathbb{Z})$) be letting $d(g\Gamma,h\Gamma)=D(g,h\Gamma)$. If $D$ ...
- 11.6k
2
votes
Accepted
Closed cobounded additive submonoid of $\mathbb{R}^n$
Yes, it has to be a subgroup. Fix $v\in M$. We need to prove that $-v\in M$. It is sufficient to find an element of $M$ arbitrarily close to $-v$.
Choose $u_1,\ldots,u_n\in \mathbb{R}^n$ so that $v,...
- 109k
2
votes
Accepted
Relationships among lattices U14, C2xG23, A15+ and their Delaunay polytopes
The answer for question (0) about $U_{14}$ is positive, as described in the Conway-Odlyzko-Sloane paper:
Let $E_8$ be the lattice in $\mathbb R^8$ with its eight co-ordinates all in $\mathbb Z$, or ...
- 9,501
2
votes
Accepted
Standard Gram matrices for lattices
How do I uniquely determine a standard Gram matrix for a lattice? Are there any other definitions of "standard Gram matrix" for lattices? Do you have any literature references in which "...
- 2,183
2
votes
Accepted
Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$
The information required for answering this question is contained in [1]. There it is scattered over many chapters; so I will give a summary here.
Let $N$ be a Niemeier lattice, $R$ be the sublattice ...
- 558
Only top scored, non community-wiki answers of a minimum length are eligible