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Questions tagged [integer-lattices]

A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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Lattice width of $conv(0,ne_1,\cdots,n e_n)$

Given a subset $K \subseteq \mathbb{R}^n$ we define the lattice width of $K$ : $$\omega(K) = \min_{d\in \mathbb{Z}^n - \{0\}} \max_{x,y \in K} d^t(x-y)$$ With $K = conv(0,ne_1,\cdots,n e_n)$ how to ...
jacopoburelli's user avatar
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Existence of a basis of lattice with successive minima norms

Is there an easy way to show that given a lattice $\Lambda \subset \mathbb{R}^n$ of full rank, exists a basis where each vector has norm $\lambda_i$ i.e the i-th successive minima ($\lambda_i(\Lambda)=...
jacopoburelli's user avatar
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Why does the lattice need to be even?

I'm studying root systems and lattices in the context of Lie algebra. In Prof. Kac's notes (https://math.mit.edu/classes/18.745/Notes/Lecture_16_Notes.pdf) it states that if $Q$ is an even (in ...
Fernando Nazario's user avatar
4 votes
1 answer
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Balanced coloring for $\mathbb{Z}^n$ with $m$ colors

Call a coloring using $m$ colors on a finite number of points in $\mathbb{Z}^n$ balanced if for any line parallel to one of $n$ axes, the difference between the number of points for any 2 colors is at ...
Dũng Nguyễn's user avatar
2 votes
1 answer
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How to compute the volume of a high-dimensional cube with complex and dependent coordinates?

I am reading a paper on ideal lattices: On page $15$ ( in the proof of Lemma $6.1$ ), it says: The cube $\mathcal{C}_2 = \left\{\left(z,\bar{z}\right) \in \mathbb{C}^{2} : \left\vert z \right\vert \...
Han's user avatar
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Shortest vector problem as hidden subgroup problem

I posted this question on the cryptography stack exchange with a bounty, but I haven’t gotten much attention. I think part of the reason might be that I’m really interested in the use of group theory ...
Joe's user avatar
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Predicting the bottom right entry in the output QR decomposition

The following may sound like a very niche question, but a good answer would advance lattice reduction. See details below. As an example, consider the matrix A = \begin{bmatrix} .111 & .025 & ....
Predrag3141's user avatar
3 votes
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Which faces does "sphere" lattice polyhedron $\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq4^a(8b+7)$ have?

$\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq 4^a(8b+7)$ is a lattice polyhedron. By Legendre's three-square theorem, such $n$ have representation(s) as the sum of $3$...
HermannSW's user avatar
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Extending a closest point to a basis for a lattice

We are given a lattice $\Lambda$ in $\mathbb{R}^2$, and we are given that it contains a basis of 2 vectors, such that the integer combinations span the whole space. Now we choose some $u$ in $\Lambda$,...
Ryan Noonan's user avatar
3 votes
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Counting sublattices of $\mathbb{Z}^n$

Let $N,j,n \in \mathbb{N}$ and let $L_{j,N}$ be the set of all sublattices $L$ of $\mathbb{Z}^n$ such that $\mathbb{Z}^n/L \cong (\mathbb{Z}/N\mathbb{Z})^j$. I want to find a formula for $\vert L_{j,...
user23571119's user avatar
1 vote
1 answer
63 views

Show $M$ is a compact surface diffeomorphic to $\Bbb T^2$

Consider the lattice $$H:=\{(e^m,e^n)\mid m,n\in\Bbb Z\}$$ which is isomorphic to $\Bbb Z^2$ via $(e^m,e^n)\leftrightarrow (m,n).$ Consider the quotient space $ M=\Bbb R^2_{\gt 0}/H $. I want to show ...
zeta space's user avatar
2 votes
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Seeking an elementry theorem about lattices.

I am doing some work with objects. Each object has a corresponding embedded free $\mathbb Z$-module, with important properties of the object being related to whether the embedding is a lattice. From ...
Sriotchilism O'Zaic's user avatar
5 votes
2 answers
123 views

No 3 vectors independent over $\mathbb{Z}$ in $\mathbb{Z}^2$, without AoC

Q: Are there three $\mathbb{Z}^2$ vectors independent over $\mathbb{Z}$ ? Context: This problem arise naturally when I'm characterizing possible sub-"latice" in $\mathbb{Z}^2$. Formally let ...
Lab's user avatar
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Proof of Blichfeldt's lemma [closed]

Blichfeldt's lemma is the following: for any full-rank lattice $\Lambda$ in $\mathbb{R}^n$, and any measurable set $D$ whose measure exceeds that of a fundamental domain of $\Lambda$, there exist $x, ...
Ari Krishna's user avatar
2 votes
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Number of lattice points of a convex body transformed by a matrix

Let $\boldsymbol{A}=[\boldsymbol{a_1},\boldsymbol{a_2},\ldots,\boldsymbol{a_T}]^\text{tr}\in\mathbb{R}^{T\times D}$, and $S$ be the integer lattice points within and on the surface of a convex body. ...
MohammadJavad Vaez's user avatar
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1 answer
35 views

Showing that the lattice D_4^+ is congruent to Z^4

I'm currently reading Conway and Sloane's Sphere Packings, Lattices and Groups. It is a great read for facts, but they only sketch proofs at a high level. Within it, they define the lattice $D_n$ to ...
psubodiosa's user avatar
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28 views

Relation between volume of a lattice and the number of integer points in this lattice

In "Algebraic Number Theory and Fermat's Last Theorem" by Stewart & Tall the volume is, as usually, defined as an integral. Now, exercise 6.1 (page 138) asks to show that the volume of a ...
3nondatur's user avatar
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(Homogeneous spaces) Moduli Space of Lattices

I'm looking at the "moduli space of $n$-dimensional lattices", which should be the double quotient $$ \mathcal{M} = \text{GL}_n(\mathbb{Z}) \backslash \text{GL}_n(\mathbb{R}) / (\mathbb{R}^\...
Johann Birnick's user avatar
1 vote
1 answer
56 views

Notion of isomorphism of lattices / intrinsic definition of lattices

I'm trying to construct the "correct" (read: a good) notion of moduli space of $n$-dimensional lattices. Here, an $n$-dimensional lattice is the $\mathbb{Z}$-span of an $\mathbb{R}$-basis of ...
Johann Birnick's user avatar
-1 votes
1 answer
345 views

Using Minkowski's Theorem to prove existence.

I need to use Minkowski’s theorem to show that if α ∈ R and Q ∈ N then there exist q, a ∈ Z such that 1 ≤ q ≤ Q and |qα − a| ≤ 1/Q. I believe that I need to define the lattice in R: Λ = {qα : 0<α&...
Ryan's user avatar
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3 votes
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110 views

What Toric Variety does this fan correspond to?

Let $\Sigma$ be the fan defined by $\{\sigma_1,\sigma_2,\sigma_3,\sigma_4,\star\}$, where $\sigma_1=\operatorname{cone}(e_1)$, $\sigma_2=\operatorname{cone}(e_2)$, $\sigma_3=-\sigma_1$, $\sigma_4=-\...
Chris's user avatar
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53 views

Lattice corresponding to the incidence matrix of a weighted tree is indecomposable

Suppose $\Gamma$ is a (necessarily connected) weighted tree with $k$ vertices $v_1,\dots,v_k$. Suppose the weight of $v_i$ is $n_i\in \Bbb Z$. Let $A$ be the $k\times k$ symmetric matrix defined by $...
user302934's user avatar
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2 votes
1 answer
75 views

What is the correct MSD function for a biased random walk on a triangular grid?

I am trying to determine the mean squared displacement $\langle r^2\rangle$ as a function of time for a discrete random walk process on a triangular grid, where each step is of size $\ell$ over a time ...
BioPhysicist's user avatar
1 vote
1 answer
34 views

Does a complex lattice determine the vectors it is generated by?

A lattice $\Lambda$ in the complex plane is defined as $$\Lambda=\{n\omega_1+m\omega_2:(n,m)\in {\mathbb Z}^2\}$$ where $\omega_1$, $\omega_2$ are complex numbers that are ${\mathbb R}$- linearly ...
Math101's user avatar
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3 votes
2 answers
53 views

Intersection of line and lattice points

Consider the Cartesian plane. A line originates from origin. How many lattice points would it cross? A lattice pointis a point with both coordinates integers. My attempt: Let the line be $y=mx$ since ...
Aarush Saharan's user avatar
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1 answer
37 views

Alternating Frobenius form in Sage

Given an alternating, non-degenerate matrix $A$ over the integers, I need to compute the matrix "closest" to the standard symplectic form that can be obtained from $A$ by an integer change ...
Oliver Miller's user avatar
1 vote
1 answer
60 views

Will the plane be filled when all the lines connecting the Lattice points are drawn?

If we draw all possible lines connecting points with integer coordinates, will the plane be completely filled? I know the answer is no, because the task of drawing lines is a countable task, while ...
زكريا حسناوي's user avatar
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61 views

Indexing lattice nodes based on distance to lattice center

Let's say there is a uniform 3D lattice of length S, where S is an odd number, and the lattice is centered at the origin. Each node has an index, but the twist is that each increment of the index must ...
Shiv-iwnl's user avatar
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doubt about volume packing lemma for intersection of convex bodies and lattices

Lemma 3.24 of Additive Combinatorics by Tao and Vu states the following: Let $\Gamma \subset \mathbb{R}^d$ be a lattice of full rank, let $V$ be a bounded open subset of $\mathbb{R}^d$, and let $P$ ...
aba's user avatar
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0 votes
1 answer
77 views

Obtaining lattice vectors for the periodicity of rectangular systems.

The network below is formed by hexagons, octagons, and squares, with its unit cell highlighted in the red rectangle. My goal is to obtain unit cells from replications of this rectangle in different ...
benjamin_ee's user avatar
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0 votes
1 answer
157 views

Counting integral coordinate points inside the region bounded by two parabolas

If we consider two parabolas $y^2=4ax$ and other one $x^2=4ay$. Both parabolas meet at origin and some other point for reference consider $(x_1,y_1)$ .Now we have to find number of integral points ...
πααρτθ Σαρθι's user avatar
4 votes
1 answer
70 views

Modeling Nanotubes Geometry

In various references, we see the construction of unit cells of carbon nanotubes (CNTs) from chiral and translational vectors. The chiral vector is given as: $$\vec C_h = n\vec a_1 + m\vec a_2$$ ...
benjamin_ee's user avatar
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0 votes
1 answer
50 views

Trying to prove that a region contains infinitely many lattice points

Let $S$ be a region in $\mathbb{R}^2$ defined to be $$ S = \{ (x, y) \in \mathbb{R}^2: \lambda_1 (x^2 + y^2)^{c} < ax + by < \lambda_2 (x^2 + y^2)^{c} \}, $$ where $\lambda_1 < \lambda_2$ ...
Johnny T.'s user avatar
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1 vote
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Uniqueness of codimension one embedding of a lattice of square-free determinant into the standard lattice

This question is a continuation of my previous question: Uniqueness of codimension one embedding into the standard lattice. Let $L$ be a positive-definite lattice of rank $n$, and suppose $L$ embeds ...
blancket's user avatar
  • 1,770
2 votes
1 answer
73 views

Bijection between group actions

I am studying Hecke operators, and one of the first claims we prove is that there exists a bijection between $\Gamma(1) \backslash \mathfrak{h}$ and $\mathbb{C}^{\times}\backslash \mathcal{L}$. Both ...
Batrachotoxin's user avatar
4 votes
2 answers
68 views

Let $S$ be a discrete set of points on a line. Does there always exist a subset of $\mathbb{Z}^2$ whose projection on the line is $S$?

I am exploring projections of the integer lattice on a line $r$ passing through the origin. I understand that the projection of a periodic 2D lattice on a line can be non periodic (a 1D quasicrystal, ...
marco trevi's user avatar
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0 votes
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119 views

What does $\mathbb{Z}_2^3$ mean? [closed]

What does $\mathbb{Z}_2^3$ mean? Is the subscript $2$ a modulo and the superscript $3$ dimensions of each element? I am studying lattice cryptography and set theory and I would like to know the how ...
smith33444's user avatar
0 votes
1 answer
49 views

Uniqueness of codimension one embedding into the standard lattice

Let $X$ be a positive definite lattice of rank $n$, and suppose $X$ embeds into the standard lattice $\Bbb Z^{n+1}$. Then is it true that the embedding is determined uniquely up to automorphism of $\...
blancket's user avatar
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5 votes
1 answer
349 views

Given unimodular matrices $A, B$, solve the matrix equation $T^\top A T = B$

Given two symmetric integer unimodular matrices $A$ and $B$ with $\det A = \det B = \pm 1$. How do we find any integer unimodular matrices $T$ such that $$ T^\top A T = B? $$ Here $T^\top$, denotes ...
zeta's user avatar
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Show explicitly the rank-24 Leech lattice that is also symmetric unimodular matrix with integer entries?

A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix: $$\begin{pmatrix} 2 &−1 &0 &0& 0& 0& 0& 0\\ −1& 2 &−1& 0& 0& 0& 0& 0\\ 0& −1&...
zeta's user avatar
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1 vote
0 answers
58 views

Matrix and Lattice Paths

I have a $k\times k$ matrix $$ A_{k}= \begin{pmatrix} 1 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & \cdots & 1 &1 & 0\\ &\vdots & &\vdots \\ 1 & 1 &...
Apple's user avatar
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1 vote
1 answer
59 views

Number of Lattice Paths with no Loops

Consider an $n$-dimensional hypercubic lattice, where the set of vertices is $\mathbb{Z}^n$. Let $E$ be the union of all edges between adjacent points (i.e. the grid lines). Given a sequence of ...
Andrew's user avatar
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1 vote
2 answers
66 views

In what sense is the geometry of the dual lattice "reciprocal" to the original lattice?

I've always struggled to get good intuition for how the geometry of the dual lattice relates to the geometry of the original lattice, even in the Euclidean case! Looking for more sources on this, I ...
stillconfused's user avatar
1 vote
0 answers
25 views

Lattice Desargues Configuration

Here's a Desargues Configuration with all points on a square lattice: Is there a more compact version, either on a square lattice or triangular lattice? Also, if there a less obvious Desargues ...
Ed Pegg's user avatar
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Integer vector subsets invariant under rational matrix

Let $A$ be an $n\times n$ matrix with rational coefficients. Define $H$ to be the maximal subset of $\mathbb{Z}^n$ such that $AH\subset H$. Question: How to describe $H$ in terms of $A$ (its Jordan ...
QMath's user avatar
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1 vote
2 answers
131 views

Properly defining a subset of $\mathbb{Z}^k$

I need to provide a good definition of the subset of $\{\{0,1,\ldots,n-1\} \times \{0,1,\ldots,n-1\} \times \cdots \times \{0,1,\ldots,n-1\}\} \subset \mathbb{Z}^k$ given by all the vertices of the ...
Marco Ripà's user avatar
  • 1,160
1 vote
1 answer
66 views

Making $\{0,\dots,9\}^2$ from smallest subset with coordinate-wise min and max

(Iranian Combinatorics Olympiad-2020) Consider the set of $100$ ordered pairs $A = \{(0, 0), (0, 1), \dots, (9, 8), (9, 9)\}$. Given any subset $S$ of $A$, we have a device to append more elements to ...
Eraser head's user avatar
3 votes
1 answer
152 views

Enclosing a lamp in space via Minkowski's theorem

I'm currently working on a problem on the chapter of Geometry and Numbers from Andreescu and Dospinescu's Problems from the Book (highly recommended to read). The problem statement is the following: ...
Ignacio Rojas's user avatar
1 vote
0 answers
19 views

Are there any six-dimensional lattices with kissing number equal to 36 or 24?

I am studying two lattices which seem to have kissing number 36 and 24, respectively, and I am curious if they are some known lattices in the literature. I have looked up Conway and Sloane but didn't ...
fagd's user avatar
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Catalan Numbers: Sequence of -1 and 1 that sums to 0 with conditions

There is a relatively simple bijection between 0-sum sequences of 1 and -1 where the sum of all partial sequences is nonnegative and Dyck paths - this is very easy to count as a Catalan number. ...
C Smith's user avatar
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