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])
664
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Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$
There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E_7$ and $\mathbb{F}_2^6 \setminus \{0\}$ (where $\...
1
vote
0
answers
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views
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 $NC$ ...
5
votes
1
answer
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Is there a relation between the number of lattice points lie within these circles
Suppose we have a circle of radius $r$ centered at the origin $(0,0)$. The number of integer lattice points within the circle, $N$, can be bounded using Gauss circle problem.
Suppose that another ...
2
votes
1
answer
513
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Kissing number lower bound vs. upper bound - precise meanings?
According to en.wikipedia.org, https://en.wikipedia.org/wiki/Kissing_number#Some_known_bounds
It says the kissing numbers $K$ have lower bound $K_L$ and upper bound $K_S$:
$$
K_L < K < K_U.
$$
I ...
8
votes
1
answer
349
views
Outer automorphisms of finitely generated linear groups
Is there an example of a finitely generated subgroup $\Gamma \subset \mathrm{GL}_n(\mathbb{C})$ such that the group of outer automorphisms $\mathrm{Out}(\Gamma)$ contains finite subgroups of unbounded ...
0
votes
1
answer
175
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Leech lattice shortest vector vs other 23 cases and E8 cases
In this paper by Viazovska, she said that:
"The E8-lattice sphere packing 𝒫E8 is the union of open Euclidean balls with centers at
the lattice points and radius $1/\sqrt{2}$." So I think ...
7
votes
1
answer
331
views
Relation between different $E_8$ matrices
There are several rank-8 square matrices known to be related to $E_8$:
Cartan $E_8$ matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Cartan_matrix
$$M_1=\left [\begin{array}{rr}
2 & -1 &...
0
votes
0
answers
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Reference on high dimensional polytopes with E8 symmetry
I'm starting to work with high dimensional polytopes. I am interested in uniform polytopes of 16-dimension and of 8-dimension (especially Elte and Gosset polytopes that have E8 symmetry).
...
0
votes
1
answer
114
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How can I find the hyperplane passing through a 600-cell
I have a 600-cell, whose coordinates are given by
$$\begin{array}{ccc}
\text{8 vertices} & \left(0,0,0,\pm1\right) & \text{all permutations,}\\
\text{16 vertices} & \frac{1}{2}\left(\pm1,\...
1
vote
1
answer
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What lattices beyond the laminated lattices (particularly in $\le 24D$) belong to a slightly expanded category that includes "descendants" of Λ13_mid?
This question is a copy of one I asked in the Math StackExchange forum a few days ago. I don't know if it qualifies as a research-level question, but it may be something beyond most people on the ...
2
votes
0
answers
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decidability special case of column generation problem
I have the following problem:
Input: sub-spaces $V_1, \dots, V_d$ of $\mathbb{Z}^{d}$
Question: are there $v_i \in V_i$ such that the matrix $(v_1, \dots, v_d)$ has determinant $\pm 1$ (equivalently, ...
1
vote
1
answer
93
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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 ...
1
vote
2
answers
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Distribution of "good" and "bad" basis in lattice families?
I'm trying to learn more about lattice based cryptosystems.
One of the fundamental ideas behind lattice based cryptosystems is that there can be many equivalent basis for a single lattice.
Formally, ...
2
votes
0
answers
202
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Modular inverse computation - avoiding Euclidean algorithm
Modular inverse is known to be computable by Extended Euclidean algorithm which is the reaping the rewards of computing the GCD of two numbers or proving two numbers are coprime.
If we already know ...
4
votes
1
answer
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The function $f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}$ for $g\in \text{SL}(2,\mathbb R)$
For $g\in \text{SL}(2,\mathbb R)$ and the Hilbert-Schmidt norm $\|\cdot\|$ (square root of sum of squares), define
$$f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}.$$
It is ...
14
votes
1
answer
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How do we know there are no more Deligne–Mostow/Thurston lattices?
In the context of hypergeometric functions, Deligne and Mostow enumerated several lattices in complex hyperbolic space/the rank 1 Lie group $\operatorname{PU}(1,n)$ (see [1] and [2]). Thurston used ...
2
votes
0
answers
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Non-uniform lattices and parabolic subgroups in Lie groups
Let $G$ be a semisimple connected Lie group and let $\Lambda < G$ be a non-uniform irreducible lattice.
How does it follows that there exists a minimal parabolic subgroup $P$ of $G$ such that the ...
4
votes
1
answer
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Inheritance of arithmeticity properties in orbifold strata
Suppose $M = K\backslash G/\Gamma$ is a quotient of a symmetric space by a lattice. I don't know all of the proper adjectives to apply here (e.g. what should be said about $G$ and so on), but I wouldn'...
3
votes
0
answers
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Torsion in the first cohomology of a lattice in a semisimple Lie group
Let $\Gamma$ be a cocompact lattice in a complex semisimple Lie group $G$ of dimension $n$. Let $M$ be a $\mathbb{Z}\Gamma$-module, finitely generated as an abelian group (let $r$ be the minimal ...
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votes
0
answers
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Are topological theta series (taking values in tmf(N)) of lattices good for anything?
I'm going to start with Mike Hopkins' great survey article in the ICM on topological modular forms (https://arxiv.org/abs/math/0212397). In it, he outlines a construction, for even unimodular lattices,...
8
votes
1
answer
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What kind of locally symmetric space is a rational sphere
Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere.
My question is the following.
Is there other ...
3
votes
0
answers
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maximizing number of lattice points with bounded diameter
Suppose I have a lattice $L$ that's just $\mathbb{Z}^k$ but scaled in every coordinate by some (potentially different) real numbers. Now I want to construct a finite set of lattice points $S \subset L$...
6
votes
5
answers
585
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Nonplanar equilateral lattice "pentagons"
It is well-known that no two-dimensional point lattice contains a regular pentagon. (See for example http://mathworld.wolfram.com/LatticePolygon.html.) The same is true for lattices in $\mathbb{R}^n$, ...
7
votes
0
answers
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K3 surfaces with no −2 curves
I seem to remember that a K3 surface with big Picard rank always
has smooth rational curves.
This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
1
vote
1
answer
187
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What information about the lattice $\Lambda$ can be recovered from the metric space $\mathbb{R}^n/\Lambda$?
Let $\Lambda\subset\mathbb{R}^n$ a lattice, i.e., a discrete subgroup that spans $\mathbb{R}^n$. Now we can look at the torus $T=\mathbb{R}^n/\Lambda$ which naturally carries the metric $d_T$ induced ...
0
votes
1
answer
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The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)
Original question (without additional information from Wendy):
Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way:
Taking the E8 as {128,...
5
votes
1
answer
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A problem of non-emptiness of intersections of certain chains of regular open sets
Let $X$ be a topological space and $\mathrm{RO}(X)$ its complete boolean algebra of regular opens. Define well inside relation: $$U\prec V\iff\overline{U}\subseteq V.$$ Let $\mathcal C\subseteq\mathrm{...
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votes
0
answers
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To show $\{(x,y) \in \mathbb Q^{\geq 0} \times \mathbb Q^{\geq 0}~:~ mn+1 \mid m^x+n^y \}$ is subset of the lattice $\{\vec u+i \vec v+j \vec w\}$?
I am writing two definitions, the $1$st one is a cover in some sense while the $2$nd one is a lattice:
Definition 1: If $m,n$ are integers bigger than $1$, then the set $$A=\{(x,y) \in \mathbb Q^{\geq ...
6
votes
1
answer
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What is the relationship between the Leech lattice and Dedekind eta function?
Like this old question, A conceptual proof of Jacobi's product formula for $\Delta$ ?, I am asking again for a conceptual proof of Jacobi's miraculous product formula for $\Delta$ (the unique ...
6
votes
1
answer
335
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Lattices in $p$-adic groups
What are the examples of lattices in $\operatorname{SL}_n(\mathbb{Q}_p)$ with $n\geq 3$ or in other semisimple $p$-adic groups of higher rank?
It is known $\operatorname{SO}_n(\mathbb{Z}[1/p])$ is a ...
8
votes
1
answer
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Siegel's Mean Value Theorem by Rogers and Macbeath
I recently became engaged in the work of Siegel, Schmidt, Rogers, Macbeath regarding random lattices and geometry of numbers, e.g. Siegel proved that
$$\int_{SL(n,\mathbb{R})/SL(n,\mathbb{Z})} \sum_{ ...
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votes
0
answers
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Mistake in Rogers' paper: "number of lattice points in a set" for the case $n=2$?
Let $f:\mathbb R^n\to \mathbb R$ be a nonnegative Borel measurable function, and
let $f^*$ be the function obtained from $f$ by spherical symmetrization (see Rogers' paper: number of lattice points in ...
1
vote
0
answers
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Second moment version of the multiple-sum Rogers integration formula
I know the following theorems due to Rogers. Let $X$ denote the space of $n$-dimensional unimodualar lattices in $\mathbb R^n$, equipped with the canonical Haar measure.
Theorem 1(Siegel-Rogers). Let ...
11
votes
2
answers
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Is there a contractible hyperbolic 3-orbifold of finite volume?
Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$.
Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that
\begin{equation}
X:=\mathbb{H}^3/\Gamma
\end{equation}
is contractible?...
5
votes
2
answers
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views
Is it still not known whether the construction of shortest nonzero vector of a lattice w.r.t. $l^2$-norm is NP-hard?
It was shown in
P. van Emde Boas, Another NP-complete partition problem and the complexity of computing short
vectors in a lattice
that the construction of a shortest nonzero vector of a Euclidean ...
3
votes
1
answer
159
views
Entire function with almost periodic boundary condition?
Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and ...
1
vote
0
answers
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Lattice packing
Let $\Lambda$ be a lattice in $R^n$ and $R>0$ a real number.
Consider the number $N$ of points in $\Lambda$ of norm less than $R$. Let $R$ goes to infinity. What can be said about the asymptotic ...
4
votes
2
answers
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How large is the set of unimodular lattices whose sucesssive minima cannot be attained by a basis of lattice?
Recall that the $i$-th successive minimum of $L\in \mathcal L$ (space of full rank lattices in $\mathbb R^d$), denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-linearly ...
2
votes
1
answer
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Proof of generalized Siegel's mean value formula in geometry of numbers
Let $\mu$ be the Haar measure defined on the space of unimodular lattices, identified with $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$.
The classical Siegel's formula in geometry of numbers states ...
1
vote
0
answers
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What is the lattice of the field $\mathbb Q_p(\sqrt[p^5-1]{p^2})$?
Let $p$ be odd prime and $\mathbb Q_p$ be the $p$-adic field. Consider the field extension $K=\mathbb Q_p(\sqrt[(p^5-1)]{p^2})$ of $\mathbb Q_p$ of degree $\frac{p^5-1}{2}$.
My question:
I want see ...
5
votes
1
answer
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Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$
To discern whether two integral quadratic forms are equivalent over the $p$-adic integers, one can compute a Jordan decomposition at $p$ and read off some invariants. Restricting to $p\ne2$ for ...
3
votes
1
answer
242
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Successive minima and the basis of lattice
I am able to prove the following two propositions: Recall that the $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-...
3
votes
0
answers
220
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Improvements to Minkowski's second theorem
Let $L$ be a (full rank) lattice in $\mathbb{R}^t$ and let $K$ be a convex body. Minkowski's second theorem states that
$$
\frac{2^t}{t!} \det(L) \leq \lambda_1 \cdot \ldots \cdot \lambda_t \text{Vol}(...
3
votes
0
answers
78
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Sums over lattice points in homogeneously expanding domains
In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $...
0
votes
0
answers
66
views
Extension of primitive set of vectors and reduction theory
Let $\Lambda$ be a unimodular lattice in $\mathbb R^d$ (unimodularity is not really necessary here but just for convenience) and let $B$ be a ball centered at the origin that contains $(k+1)$-many $\...
1
vote
0
answers
58
views
Finding a particular kind of basis of subgroup of a lattice generated by non-negative part
For $\mathbf v=(v_1,\ldots,v_n)\in \mathbb Z^n$, let $\operatorname{supp}(\mathbf v):=\{j: v_j \ne 0\}$. For a subset $X$ of $\mathbb Z^n$, define $\operatorname{supp}(X):=\bigcup_{\mathbf v \in X} \...
2
votes
0
answers
132
views
Lattice relations and isogenous elliptic curves
Consider two (primitive) elements $\pi_{i} \in \mathbb{C}$, such that $\pi_{1} = M \pi_{2}$ for $M \in \mathcal{S}_{m}$ with $$\mathcal{S}_{m}:=\Big\{\begin{pmatrix}
A & B \\
0 & D
\end{...
1
vote
0
answers
41
views
Barnes-Wall lattices’ contact polytopes
The contact polytopes of the Barnes-Wall lattices in 1, 2, 4, and 8 dimensions are all uniform polytopes. Is this true in any higher number of dimensions?
4
votes
0
answers
122
views
Lattice reduction of basis with non-integer coefficients
Suppose I have an ordered basis $\{b_1, \dots, b_n\}$ of a lattice in $\mathbb{R}^n$, but I do not assume that $b_i \in \mathbb{Z}^n$ for all $1 \leq i \leq n$.
I would like to perform lattice ...
2
votes
0
answers
105
views
Find the closest point of a lattice $\Lambda$ given the closest point for its union of cosets $\bigcup_i ({\bf r}_i+\Lambda)$
Suppose we have an $n$-dimensional lattice $\Lambda$, and a set of vectors ${\bf r}_i$, we can construct a union of cosets of $\Lambda$, denoted as $L$, as
$$
L \equiv \bigcup_i ({\bf r}_i+\Lambda)
$$
...