# 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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### Polynomial-time quantum algorithms for lattice problems (GapSVP, SIVP, LWE)

The author of a recent preprint claims to have found polynomial-time quantum algorithms for solving the following lattice problems: the Decisional Shortest Vector Problem (GapSVP), the Shortest ...

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### Why do symmetries of K3 surfaces lie in the Mathieu group $M_{24}$?

I'm having trouble following some steps of this argument from the appendix of Eguchi, Ooguri and Tachikawa's paper Notes on the K3 surface and the Mathieu group M24:
Now let us recall that the ...

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### Arithmetic lattices are finitely presented

In the book "Kazhdan's Property (T)" by Bekka-de la Harpe-Valette, the following is stated on p.6 of the introduction:
"Of course, it is classical that arithmetic lattices are finitely ...

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### Seeking Article "Generating random lattices according to the invariant distribution" by M. Ajtai

I am searching for a specific article titled "Generating random lattices according to the invariant distribution" authored by Ajtai. Despite being widely cited in various papers, I have been ...

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### Integral representation of completely alternating homogeneous functionals on semi-lattice of continuous functions

For a long time I've been interested in G. Choquet seminal work "Theory of capacities" (Annales de l’institut Fourier, tome 5 (1954), p. 131-295). More precisely part 53 about integral ...

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### On shortest vector problem

Assume we have an oracle which gives the length of the shortest vector in a lattice. Given this oracle can we find the shortest vector in polynomial time?

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### What is the smallest $\mathbb{Z}[x]$ multiple of $(x-1)^n$ in the coefficient vector $\ell_1$ sense?

Define a norm $\lVert p \rVert_1$ for $p\in \mathbb{Z}[x]$ as the sum of absolute values of the coefficients of $p$, as expressed in the ordinary monomial basis. What is the smallest norm of a ...

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### Does $\mathbb Z^n$ contain $A_n$?

Are there any positive integer $n > 3$ such that the root lattice $A_n$ is contained in $\mathbb Z^n$?

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### Vanishing of $\ell^2$-Betti numbers of $\mathrm{GL}(n,\mathbb{Z})$ for $n\geq 3$

$\DeclareMathOperator\GL{GL}$In a paper I read the following claim:
By the work of Borel the $\ell^2$-Betti numbers of the cocompact lattices of $\GL(n,\mathbb{R})$ are known to all vanish when $n ≥ 3$...

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### Kripke frame, lattice and some intermediate logics

For a given finite and rooted intuitionistic Kripke frame $\mathcal{F}$, let $\log(\mathcal{F})=\{\phi : \mathcal{F}\vDash \phi\}$ and assume $S=\{\log(\mathcal{F}): \mathcal{F} \text{ is finite and ...

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### Lattice generated by parabolics

Let $G$ be a semisimple Lie group of split-rank one and let $\Gamma$ be a non-cocompact lattice which is torsion-free.
For the group $G=\mathrm{SL}_2(\mathbb{R})$ it then follows that $\Gamma$ is ...

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### What is the lattice point distribution over binary quadratic forms?

Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$.
For simplicity, we keep things only on quadrant I of the ...

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### Lattice not contained in any connected subgroup is not contained in any positive dimensional subgroup

Let $ G $ be a simple Lie group and let $ \Gamma $ be a lattice in $ G $. If $ \Gamma $ is not contained in any connected subgroup of $ G $ does that imply that $ \Gamma $ is not contained in any ...

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### Kirszbraun-like extension of periodic functions

Let $\Lambda \subset \Lambda' \subset \mathbb{R}^n$ be lattices. Let $f : \Lambda' \rightarrow \mathcal{H}$ be a $a$-Lipschitz function, where $\mathcal{H}$ is a finite-dimensional Hilbert Space. ...

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### Roots in indefinite lattice of K3 surfaces

Anyone who likes $K3$ surfaces cares about lattices of the form $$ (2d)\cdot y^2 - 2x \cdot z$$ (namely the mukai pairing on $H^*_{alg}(K3)$ of picard $1$ with polarization $d$).
Inside we have ...

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### Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?

Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...

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### Efficient decoding of the E8/Leech lattice

Background:
Our goal is to quantize a sequence of floating point numbers generated i.i.d. from a standard Gaussian source and minimize the MSE reconstruction error. We can use two bits for each sample....

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### Number of points in a ball in positive characteristic

Let $w_1,\cdots,w_n$ be of elements of $\Omega$, that is the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by the $-\deg$ valuation.
Assume that $w_1,...

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### Lattices and noncommutative algebras in noncommutative geometry

This a question that I've asked in mathematics stack exchange without having received any response :
I am interested in the relation between lattices and noncommutative algebras in the context of ...

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### How common is it that the number of the shortest vectors in a lattice is exactly two?

The lattice $\Gamma$ in $\mathbf{R}^{m}$ with the lattice basis $\{ke_{k}\}_{k=1}^{m}$ has exactly two shortest vectors: $\pm e_{1}$.
My question is the following:
Among all the lattices with fixed ...

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### Automorphism group of a Lorentzian lattice

Consider the even integral lattices $L_n:=Z\times Z\times Z^{n-2}$ (where $Z$ is the set of integers) with elements $x=(x_+,x_-,x_d)$ and inner product
$$(x,y):=x_+y_-+x_-y_++2x_d\cdot y_d.$$
Its ...

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### Exponential growth of shortest vector norm for successive lattices corresponding to powers of a matrix

Let $A\in M_{2\times 2}(\mathbb{Z}) $ be a two by two integer matrix such that $0,\pm 1$ are not eigenvalues of $A$ and $\left|\det(A)\right|>1$. I am interested in the growth of the norm shortest ...

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### Expected value of the length of the shortest non-zero vector in a lattice?

$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(...

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### Will Coppersmith's method work for this bivariate modular polynomial shape?

I have a bivariate modular polynomial of shape
$$f(x,y)=x^2y-g(x)\equiv 0\bmod q$$
where
$q=(2p-1)(2p+1)$ is a product of two primes $2p-1$ and $2p+1$,
$g(x)\in\mathbb Z[x]$ is of degree four and
$f(...

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### Where has this structure been observed?

$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure:
$R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation":
$$R_X (x, y) \cdot R_Y (x +...

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### Advice on results for balls on regular $N$-dimensional grids

I have obtained some results regarding balls on regular $N$-dimensional grids. I would like expert opinion on wether the results are significant or interesting enough for (trying to) publish them in a ...

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### Condition on the minimality of Minkowski units

I am interested in to undrestand when the Minkowski units in real biquadratic number fields are minimal in the log unit lattices.
I have read some pieces of literature online which are investigating ...

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### Characterizing the D4 lattice as a sphere packing

Suppose I pack spheres in $\mathbb{R}^4$ in such a way that each touches 24 others. (All spheres in my question are assumed to have equal radius and be non-overlapping.) Does this packing ...

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### About finitely generated lattices in Lie groups

Let $G$ be a connected Lie group. Let $\Gamma$ a lattice in $G$ not necessarily uniform (cocompact). Is it true that $\Gamma$ is finitely generated?

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### What can lattices tell us about lattices?

A general group-theoretic lattice is usually defined as something like
A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...

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### Genus of quadratic form

I am trying to understand the genus of a lattice from Conway and Sloane textbook. They said two quadratic forms $Q_1$ and $Q_2$ lie in the same genus if they are equivalent over $\mathbb{R}$ and over ...

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### group generated by unipotents in arithmetic subgroup is finitely generated

Let $G$ be a semisimple algebraic $\mathbb{Q}$-group and $\Gamma$ an arithmetic subgroup of $G$. In particular $\Gamma$ is finitely generated.
Denote by $\Gamma^{u}$ the set of unipotent elements in $\...

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### The sum of $q^{-2}$ over nonzero Gaussian integers

I'm reading about the Weierstrass zeta function. In this context,
$\phi(z)=\zeta(z)-\pi\bar{z}$
is periodic over the lattice
$$\mathcal{L}=\{a+bi\mid a,b\in\mathbb{Z}\}.$$
If we take $w\in\mathcal{L}\...

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### Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?

$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...

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### Preserve validity between the two Kripke frames

The background of our discussion is intuitionistic logic, i.e. the following definitions are intuitionistic Kripke frame.
For $n \geq 1$, let $\mathcal{C}_n$ denote the frame which is shown in Fig.1. ...

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### Extremal problem for 2-dimensional lattices

Given a lattice $L$ in a Banach space $(B,\|\;\|)$, one denotes by $\lambda_1(L)$ the least norm of a nonzero element in $L$, and by $\lambda_k$ the least $\lambda$ such that there is a linearly ...

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### Cocompact lattices in $\mathrm{Sp}(n, 1)$

This is a continuation from my previous question. I am reading the following paper of Cowling-Haagerup, and I was wondering whether there are uniform lattices in $\mathrm{Sp}(n, 1)$. Is there some way ...

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### Questions related to a paper of Cowling-Haagerup and uniform lattices of $\mathrm{Sp}(1,n)$

I am reading the following paper of Cowling and Haagerup for my master’s thesis. I am new to this area so I am not very conversant. So I do apologize if the questions are silly.
Question 1. In the ...

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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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### 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 ...

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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 ...

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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 ...

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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).
...

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### 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 &...

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### A note on orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$.
...

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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, ...

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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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### Cohomology of cocompact lattices in hyperbolic spaces

I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...

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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 ...

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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 ...