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.

learn more… | top users | synonyms

3
votes
1answer
342 views

Generators of Special Linear Groups

Linear algebra and special-linear group experts please help: I learn that in principle one can generate this $M$ matrix form the $B_1$ and $B_2$ matrix below. Here $$ M=\begin{pmatrix} 0& 1& ...
7
votes
1answer
65 views

Orbits of action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$

I'm considering the action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$: if $A\in SL_m(\mathbb{Z})$ and $v\in\mathbb{Z}^m$, then $Av\in\mathbb{Z}^m$. My question is: what are the orbits of this action? ...
0
votes
0answers
56 views

Which of the following sets are sub lattices of $\mathbb{Z}^2$?

Here are the first three sets: $\{(x, y) \in \mathbb Z^2 : x + y = 1\}$. $\{(x, y) \in Z^2 : x + y = 0\} = S^2$. $\{(x,y) \in Z^2 :2\mid x\} = S^3$ I found that the first one is not a subgroup ...
1
vote
1answer
87 views

Obtaining a new basis for a lattice with one of the new basis vectors fixed

Suppose that a lattice $L$ in $\mathbb{R}^3$ is given with a basis $B = \mathbf{ \{ v_1, v_2, v_3 \} }$. Is there an algorithm that would help me obtain a new basis $B' = \mathbf{ \{ v_1', v_2', v_3' ...
1
vote
1answer
51 views

Orbits of action of $SL_2(\mathbb{Z})$ on lattice

I'm interested in the action of $SL_2(\mathbb{Z})$ on $\mathbb{Z}^2$: if $A\in SL_2(\mathbb{Z})$ and $v\in\mathbb{Z}^2$, then $Av\in\mathbb{Z}^2$. Specifically, what are the orbits of this action?
1
vote
0answers
28 views

Which lattices are ideals of a number field?

Let $K$ be a number field, then its ring of integers $\mathcal{O}_K$ in the Minkowski space of $K$ is a lattice $\Lambda$. Is there some geometric descrpition/intuition that describes sublattices of ...
3
votes
0answers
40 views

Number of points in the diagonal of an $X \times Y$ square lattice box.

So assuming we have a $X \times Y$ lattice, Say for example a $3 \times 5$ like so * * * * * * * * * * * * * * * I need to find the number of points that each ...
1
vote
0answers
31 views

Non zero Unimodular matrices

I have a question about the lattice reduction algorithms like LLL algorithm. Lattice reduction algorithms like LLL generate a unimodular matrix which makes more orthogonal basises for a given matrix. ...
4
votes
1answer
75 views

Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to ...
1
vote
1answer
64 views

Factoring an integer matrix into unimodular and upper triangular matrices

I'm stuck at the following problem. I've read this somewhere, but the author did not provide a proof, probably assuming that this is `clear'. Let $A \in \mathbb{Z}^{n \times n}$. Then, there exist a ...
1
vote
0answers
37 views

Is the Voronoi region of a lattice symmetric around origion?

Assume an n-dimensional lattice. Is the Voronoi region of the lattice symmetric around origion? In other word, is the following statement true? "if $x\in \mathcal{V}$ then $-x\in \mathcal{V}$" where ...
1
vote
1answer
120 views

Lattice definition as finite density infinite set of vectors closed under addition?

This question came up in the definition of lattices (in the crystallography/group theory sense, not the ordered set sense) in our condensed matter lectures, but I believe it's more appropriate here ...
1
vote
1answer
47 views

Weyl group and weight lattice chambers.

Consider two simple Lie groups $G_1$ and $G_2$. Let $G_1$ have $W_1$ as a Weyl group and $G_2$ have $W_2$ as a Weyl group. Is it true that the Weyl group of $G_1 \times G_2$ is $W_1 \times W_2$? ...
1
vote
0answers
55 views

Closest vector problem

Given is a vector $v=\begin{pmatrix}2,&-1,&0,&1\end{pmatrix}$ as the shortest vector of the lattice $\Lambda (B)$, where $B$ is determined as $B=\begin{pmatrix}4 &-3 & 2 & 0\\ ...
1
vote
1answer
25 views

Continuity of the mapping of a lattice in $\mathbb{R}^{d}$ onto the length of the shortest vector

I have the following problem: Let $X_{d}:=\{\mathbb{Z}^{d}g;g\in\operatorname{SL}_{d}(\mathbb{R})\}$ and define $\lambda:X_{d}\to(0,\infty)$ by: $$ \lambda(\Lambda):=\min\left\{r>0;\Lambda\cap ...
1
vote
1answer
31 views

isotropic sublattice

In Scattone " Compactification of Moduli Spaces of Algebraic K3 Surfaces" the author cites a correspondance between primitive isotropic sublattices of some lattice L and the rational boundary ...
0
votes
1answer
50 views

Equivalence of Lattices

Let $\Gamma=\{mw_1+nw_2:m,n\in\mathbb{Z}\}$ and $\Gamma'=\{mw_1'+nw_2':m,n\in\mathbb{Z}\}$. Show that $\Gamma=\Gamma'$ if and only if there exists a matrix $A\in SL(2,\mathbb{Z})$ such that $\left( ...
1
vote
0answers
63 views

Which matrices generate the same lattices?

Given are the following matrices: $A_{1}=\begin{bmatrix} 4 & 7 &9 \\ 3 & 5 &3 \\ 1 &2 &0 \end{bmatrix}$ , $A_{2}=\begin{bmatrix} 10 & 9 &6 \\ 11 & 7 &2 \\ ...
0
votes
1answer
83 views

Show that $(\Bbb{N}, |)$ is a distributive lattice.

Show that the set of Natural numbers with divisibility form a distributive Lattice where for any $x, y\in\mathbb{N}$ we have $x$ meet $y = \operatorname{gcd}(x,y)$ and $x$ joint ...
1
vote
1answer
222 views

How to count lattice points on a line.

How can we count the number of lattice point on a line, given that the endpoints of the lines are themselves lattice points? I really can't think of how counting lattice points would work, so please ...
5
votes
1answer
487 views

Shortest Non-Zero Vector in Integer Lattices with Given Points

There are two questions related to the shortest non zero vector problem that have left me scratching my head. Please bear with me as I describe the problem. Disclaimer: this is homework. For the ...
12
votes
2answers
462 views

Show $\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$ by block-walking interpretation of Pascal's triangle

A combinatorial proof for the identity $$\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$$ is the following "committee" argument, which seems the most common. There are $\binom{n}{k}$ ...
1
vote
0answers
33 views

integer programming with bounded dimension

We know that integer programming with bounded dimension or fixed number of variables can be solved in polynomial time by Lenstra's result(from results of the LLL algorithm). After heavy foraging i ...
3
votes
0answers
50 views

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 $\mathbb{R}^n$, ...
2
votes
1answer
58 views

Lattice points in spheres

Let $\mathbb{R}^n$ have the standard Euclidean metric and call a point $P = (x_1, \ldots,x_n)\in\mathbb{R}^n$ a lattice point if for all $i$, $x_i\in\mathbb{Z}$. Allowing small number theoretic ...
1
vote
1answer
35 views

What is a good introduction to quantities such as the norm of a lattice and of short vectors in the context of lattice reduction?

I am trying to make sense of different notations used in measuring lattices, in particular before and after a basis reduction. In particular, I am trying to get bounds and estimates for the size of ...
2
votes
0answers
127 views

Lattice theory in mathematics and physics

I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in ...
1
vote
0answers
38 views

Inclusion-minimality of a lattice basis

An integer lattice is a subgroup of $\mathbb{Z}^n$. Since $\mathbb{Z}$ is PID, each lattice has a well-defined rank and a generating set of rank many elements is a basis. I wonder if there is a way ...
2
votes
1answer
50 views

Question regarding polygons

Can you prove, that if a equilateral lattice n-gon is constructible, then there will be such a polygon for which the sides have minimal length?
3
votes
1answer
106 views

Bounding the density of finite coprime sets

I am currently running into a problem related to coprime numbers. Consider a set of $d$-dimensional integer vectors, $z \subset \mathbb{Z}^d$ such that each component $z_i$ is bounded by another ...
0
votes
1answer
142 views

Lattice Squares; Basic Interesting Facts and Problems

I'm going to write an article in an educational magazine for middle school students, about the game Square It. The purpose of the game is to make lattice squares: I want to introduce the game, and ...
1
vote
0answers
87 views

A lattice-theoretic question related to noncommutative tori

[NCG] So I'm trying to pin down a fairly well-known bit of noncommutative-geometric folklore that says that for $\Theta \in M_N(\mathbb{Q})$ skew-symmetric, the corresponding noncommutative $N$-torus ...
2
votes
1answer
146 views

Proof of the finite number of Bravais lattices?

I've been taught that there are a finite number of Bravais lattices in 1, 2 and 3 dimensions. I am wondering if there is a proof of this fact. Maybe this is obvious and I am only missing certain key ...
2
votes
1answer
1k views

Find point on line that has integer coordinates

Given a 2D line e.g. (3,10) -> (8.3,16.5), how can I find any point on that line that has has whole-number coordinates? I can easily iteratively walk along one ...
0
votes
1answer
192 views

Lattices in the complex plane

Consider the ring $R=\mathbb{Z}[\sqrt{-2}]$. It is a lattice in the complex plane: the set of points with integer coordinates with respect to the basis: $1,\sqrt{2}i$. Each mesh of the lattice is a ...
2
votes
2answers
90 views

Find lattice points on a planar curve

I have the following curve in the plane: $$y = \frac{c-x}{6x+1}$$ Given a constant value $c \in \Bbb N$; is there a technique(s) I can apply to find lattice points on this curve?
0
votes
1answer
22 views

What's the maximum number of sums of $ak_1,..,ak_m, b\ell_1,…,b\ell_m$ needed to solve for $a$ or $b$?

Given a set of integer multiples of $a$ and $b$, $ak_1,..,ak_m, b\ell_1,...,b\ell_m$, what is the maximum number of finite sums of the multiples you can create such that no sum of all multiples of $a$ ...
4
votes
2answers
69 views

Looking for references on 'non-discrete lattices'

A lattice in $\mathbb{R}^n$ is a discrete subgroup that spans $\mathbb{R}^n$. Recently I've been running into a similar sort of object consisting of more than $n$ vectors in $\mathbb{R}^n$ and their ...
4
votes
0answers
121 views

Adelic lattices

Let $\mathbb{A} = \widehat{\mathbb{Z}} \otimes \mathbb{Q} \times \mathbb{R}$ be the adeles over $\mathbb{Q}$. In Deligne's article "Formes modulaires et representations de GL(2)" he states without ...
1
vote
0answers
64 views

Why the following observations regarding lattices hold?

The following is an excerpt of a recent paper on lattice cryptography: Let $n$ and $q$ be integers [...], and let $\beta > 0$ . Given a uniformly random matrix $A \in \mathbb Z^{n \times m}_q$ ...
1
vote
2answers
435 views

When is the number of lattice paths from $(1, 1) \to (x, y)$ divisible by $3$?

Let $S$ be the set of $\{(1,1), (1,−1), (−1,1), (1,0), (0,1)\}$-lattice paths which begin at $(1,1),$ do not use the same vertex twice, and never touch either the $x$-axis or the $y$-axis. Let ...
2
votes
2answers
124 views

Name for grid system

Is there a name for a type of grid you might find in Battleship? Where coordinates don't relate to points on a grid but rather the squares themselves?
1
vote
1answer
136 views

Fundamental volume of quotient group

I came across this rule in my old notes, but I need an explanation to how it could possibly originate: The theorem says that for any lattice $L$ in $\mathbb{R}^n$, the order of the quotient group, ...
3
votes
1answer
91 views

Basis reduction and continued fractions

While reading several articles about lattice basis reduction I am left with a few questions. For one, I came across this piece of text Let $\alpha$ and $\beta \in \mathbb{R}$. Then there are two ...
0
votes
0answers
174 views

Its just one point… How do I find it?

Okay so here is the deal... I have a CLOSED convex polyhedron $Ax \le b$ (where $x$ is in $R^n$) and it has i vertices denoted $V_i$ such that $V_i = (x_{i1}, x_{i2}, \ldots, x_{iN})$ where $0 \le ...
2
votes
1answer
71 views

Lattices in $\mathbb{C}$

I have the following assignment: consider the map $$|\cdot|:\mathbb{Z}[i]\longrightarrow \mathbb{N},\qquad |a+ib|:=a^2+b^2$$ 1) Prove that $|\alpha|<|\beta|$ iff $|\alpha|\leq |\beta|-1$ and ...
0
votes
1answer
168 views

Mapping Between Sequences: Example

Take $0\leq r < m$, and let all values be nonnegative and integer. Consider the function on a sequence ${x(n)}$, $\Phi_mx(mn+r)=mx(n)+\frac{r}{m}(x(n+1)-x(n))$, where we consider $x(0)=0$. As an ...
2
votes
3answers
46 views

Is it possible to make a regular 3-polygon by selecting $3$ points in $S$

I would appreciate if somebody could help me with the following problem: Let $\mathbb{Z}$ be the set of all integers and let $ S = \mathbb{Z} \times \mathbb{Z} $. Question: 1). Is it possible to ...
7
votes
1answer
119 views

What is the limit $\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le m/n\le r/s}x^my^n$?

Let $S=[0,1)^2$ and $m,n$ are positive integers and $p/q,r/s$ are positive rationals with $p/q<r/s$. What is the limit $$\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le ...
1
vote
1answer
252 views

Does “short integer solution” lattice problem admit hard instances with q=2?

Let $q$ be a prime, $m,n$ be integers with $m>n$, and $\beta$ be a real number. Moreover, let $A$ be a matrix in $\mathbb Z^{n \times m}_q$. In the "short integer solution" (SIS) lattice problem, ...