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

7
votes
1answer
118 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
190 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, ...
1
vote
1answer
168 views

How tell if a polyhedron contains a lattice point

So given a polyhedron $Ax \le b$ Is there an Algorithm or formula to determine whether said polyhedron contains a lattice point (integer point) I was thinking a couple things: brute force ...
1
vote
2answers
581 views

Monotonic Lattice Paths and Catalan numbers

Can someone give me a cleaner and better explained proof that the number of monotonic paths in an $n\times n$ lattice is given by ${2n\choose n} - {2n\choose n+1}$ than Wikipedia I do not understand ...
0
votes
0answers
57 views

Determinant of a Lattice Code

For a binary $[n,k,d]$ code $C$ with lattice $\Lambda (C)$ I want to show that $det(\Lambda (C)) = 2^{n-2k}$. I'm having a little trouble with demonstrating this and it would help me very much ...
0
votes
1answer
81 views

Generating vectors of the face-centered cubic lattice

I can't find a reference for a set of generating vectors for the Tetrahedral-octahedral honeycomb lattice. I would like to know the "canonical" set and if possible a more general set described by ...
9
votes
2answers
513 views

Generalizing the 290 theorem.

I have only just come across the remarkable theorem of Conway about universal quadratic forms over $\mathbb{Z}$; namely that in determining whether a integer coefficient, positive definite quadratic ...
2
votes
2answers
93 views

What is a good way to simplicize the integer lattice?

I have a function $f$ defined on the positive integer lattice in $\mathbb{R}^n$ (i.e. the vectors in $\mathbb{R}^n$ whose coordinates are all nonnegative integers). I want to extend the domain of $f$ ...
1
vote
5answers
84 views

Represent any number with two factorials

I was wondering if it is possible to represent any positive integer with x! - y! ? If not, is there any proof?
2
votes
0answers
43 views

When does a simplex have an interior lattice point?

Given $r$ vectors $v_1, \dots, v_r$ in $\mathbb{Z}^n$, is there an easy way (in terms of the entries of the $v_i$) to determine if there is a point of $\mathbb{Z}^n$ in the interior of the simplex ...
0
votes
2answers
170 views

Counting lattice points interior to a polygon

If I define an integer lattice $\Lambda \subseteq \mathbb{Z}^2$ with a basis given by $$\omega_{1} = a \hat{i} + b\hat{j}, \;\;\; \omega_{2} = -b \hat{i} + a\hat{j}$$ How can I count how many lattice ...
1
vote
1answer
46 views

How can I tell if two points projected onto a tiled hyperplane are in the same tile?

I have a hyperplane defined by two vectors in $\mathbb{Z}^3$ and I have tiled the hyperplane by parallelograms defined by the two vectors. Then I have two lattice points that I want to project down ...
0
votes
1answer
44 views

Lattice formed by a linear congruence

Let there be a linear congruence $a+b y \equiv 0 \pmod{m}$, with $y$ and $m$ ($m$ is a prime) values known. Do all the integer $(a,b)$ pairs satisfying the congruence form a lattice? If yes, how can I ...
0
votes
0answers
42 views

Determinant of a superlattice

I'm beginning to study lattices and cannot solve the following exercise from the book: Let $\Lambda=\left\langle\mathbb{Z}^n,\left(\frac{a_1}q,\ldots,\frac{a_n}q\right)\right\rangle$, where ...
1
vote
1answer
59 views

Basis condition in lattices

Let $\Lambda$ be a lattice in $\mathbb{R}^n$ (i.e. a discrete subgroup spanning the whole space). Given a basis $e_1,\dots,e_n$ of the lattice, we can consider the fundamental parallelogram $P$ ...
2
votes
2answers
3k views

Lattice paths and Catalan Numbers

Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner. How many such routes are there through a ...
4
votes
2answers
211 views

Guaranteeing an integer lattice point centroid

My question is this: Writing $n(4)$ to be the minimum number of integer lattice points in the plane so that some four of them must determine an integer lattice point centroid, show that $n(4)=13$. I ...
0
votes
0answers
98 views

Counting lattice points in a square region

We have a lattice which is spanned by the vectors $(a,b)$ and $(-b,a)$, where $a,b \in \mathbb{N}$. Now we have a square region centered at $(0,0)$ having sides of length $2r+1$ for $r\in \mathbb{N}$. ...
4
votes
1answer
152 views

What is the easiest way to describe the Leech lattice explicitly?

I am aware that the Leech lattice is the unique even unimodular lattice in $\mathbb{R}^{24}$ with no norm $2$ vectors. However I am after a way to describe this lattice explicitly without reference ...
1
vote
1answer
203 views

What happens when the basis vectors of an integer lattice are not linearly independent?

The definition of a lattice requires basis vectors that are not linearly independent. Why? Specifically the following three vectors are linearly independent and form the basis of a lattice: ...
6
votes
1answer
120 views

Power Functions on the Integers

Suppose $f:\mathbb{R}\to\mathbb{R}$ is of the form $f(x)=x^a$ for some $a\in\mathbb{R}^{+}$. If $f(\mathbb{Z})\subset\mathbb{Z}$, show that $a\in\mathbb{Z}$. Source: A friend posed this problem; not ...
0
votes
0answers
62 views

Orthogonality in a lattice

Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19). Let $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda_{\mathbb{R}}$ a positive subspace of dimention 3. ...
2
votes
1answer
33 views

dist($av$, $L$) = $a \cdot $ dist($v$, $L$)?

If $L$ is a lattice subspace and $v$ is the closest lattice point to $L$, why does dist($av$, $L$) = $a \cdot $ dist($v$, $L$)? I saw this step in a proof that every lattice has a lattice basis. (I ...
5
votes
1answer
189 views

Bézout's identity in higher dimensions?

I have an invertible rational matrix $C\in\text{GL}(n,\mathbb{Q})$ which works on lattice $\mathbb{Z}^{n}$. Can I write the resulting set in the following form $$C\cdot \mathbb{Z}^{n}=X\cdot ...
2
votes
2answers
194 views

A problem involving the lattice grid.

Suppose that $22$ points are arbitrarily chosen from a $7\times 7$ lattice grid. We are to prove that there exists at least one rectangle in any $4$ points chosen from the above $22$. A general ...
15
votes
5answers
736 views

Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$

Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec ...
1
vote
1answer
229 views

How do you construct a lattice from its basis or its Gram Matrix?

I'm really having trouble trying to understand this. A few weeks back, I got pretty interested in sphere packing and I'm trying to grasp the idea of using a matrix to represent the basis of a lattice. ...
3
votes
1answer
143 views

Classifying all ideals of a lattice $\mathbb{Z}[\sqrt{-d}]$

In Artin's Algebra he presents a method (that I am sure I am butchering) for classifying ideals of a given lattice $\mathbb{Z}[\sqrt{-d}]$ by taking any ideal $I$, choosing an element of minimum norm ...
0
votes
1answer
65 views

Lattice generated by elements of minimal length

I'm following Miranda's book on algebraic curves and Riemann surfaces. In the section where he talks about the automorphisms group of the complex tori he claims the following: Let $L$ be a lattice in ...
2
votes
1answer
117 views

Planar cross section of leech lattice?

I was wondering what any planar cross sections of the leech lattice would look like. I don't know much about this topic at all, I'm just quite curious. Is there any way to find out?
2
votes
0answers
48 views

Automorphisms of a lattice and changing to a nicer $\mathbb{Z}$-base

Suppose I have an integral lattice $L$ with an arbitrary $\mathbb{Z}$-base, equipped with a positive-definite nondegenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$, and an isometry $\nu$ ...
0
votes
2answers
202 views

Finding a basis for a complex lattice given a nondivisible vector in the lattice

If I am given some lattice defined as, say $$L=\{Az_1+Bz_2\ |\ A,B \in\mathbb{Z}\}$$ and a vector $v=az_1+bz_2$ , where $\gcd(a,b)=1$, I would like to find another vector $\,w\in L\,$ such that ...
2
votes
0answers
117 views

Lattice Reduction Problem: Minimizing the “Longest” Basis Vector

Suppose we have a basis for an integer lattice formed by the vectors $\vec v_1, \vec v_2, \ldots,\vec v_n$. Then let $A$ be the augmented matrix $( \vec v_1| \space \vec v_2| \cdots |\space \vec ...
1
vote
1answer
90 views

Solution count of quadratic form congruence over $\Bbb Z / 8 \Bbb Z$

Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
0
votes
1answer
445 views

Index of a sublattice in a lattice and a homomorphism between them

I am asked to show that if $\phi_A$ is the homomorphism from $\mathbb{Z}^k \rightarrow \mathbb{Z}^k$ given by $\phi_A(x)=xA$ then the index of $\phi(\mathbb{Z}^k)$ in $\mathbb{Z}^k$ is finite if ...
1
vote
0answers
134 views

solution count of quadratic form congruences

Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
3
votes
1answer
129 views

Closest vector problem for orthogonal lattices

Let's say I have a reduced basis $\mathcal{B}$ for an orthogonal lattice in $\mathbb{R}^n$, then the Shortest Vector Problem is trivial (the shortest vector in the basis). According to my intuition, ...
3
votes
1answer
128 views

Bounds for the size of a circle with a fixed number of integer points

I know that there are infinitely many rational points on the (unit) circle. I am interested in the following question: How large has the radius of a circle to be, such that there are at least $n$ ...
2
votes
0answers
51 views

A certain labeling of lattice points on the plane [duplicate]

Possible Duplicate: A stronger version of discrete “Liouville’s theorem” Let each lattice point of the plane be labeled by a positive real number . Each of these numbers is the arithmetic ...
3
votes
2answers
292 views

Number of lattice points

Does there exist a formula for counting the number of lattice points not outside of a square , with the at most information available concerning the square are the position coordinates of the four ...
0
votes
1answer
337 views

Proving Binomial Identities Using Bijections To Lattice Paths

How can I derive a bijection to show that the following equality holds? $2\displaystyle\sum\limits_{j=0}^{n-1} \binom{n-1+j}{j} = \binom{2n}{n}$ In class, we've been deriving bijections using ...
3
votes
1answer
287 views

Good textbooks for lattice and coding theory

I am looking for good textbooks for lattice and coding theory. Lattice and coding theory are very interesting on their own, but I have application of the theory to K3 surfaces & modular forms (and ...
1
vote
1answer
42 views

Multiplications by unimodular matrices

I feel like this must have an obvious answer, but my knowledge of integer arithmetic is limited. Given an (integer) matrix $A$ of dimension $m \times n$ and an unimodular matrix $U_l$ of dimension $m ...
2
votes
0answers
68 views

Term for intersection of lattice and convex region?

Is there a special term or convenient phrase for the restriction of a convex region to points of a lattice? This is motivated by wanting to talk about the feasible points of a discrete problem. I'd ...
4
votes
1answer
167 views

Is there an algorithm to find a basis for the lattice $V \cap \Bbb{Z}^n$ given a basis for $V \subseteq \Bbb{Q}^n$?

This might be a stupid/very simple question, but since I can't quite seem to come up with a nice trick I will ask it anyway. Assume that we have a vectorspace $V \subseteq \mathbb{Q}^n$ given in the ...
2
votes
2answers
673 views

Finding a basis for the solution space of a system of Diophantine equations

Let $m$, $n$, and $q$ be positive integers, with $m \ge n$. Let $\mathbf{A} \in \mathbb{Z}^{n \times m}_q$ be a matrix. Consider the following set: $S = \big\{ \mathbf{y} \in \mathbb{Z}^m \mid ...
6
votes
0answers
214 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
2
votes
0answers
234 views

Probability distribution of a self avoiding walk

Preliminary: Consider a walk on the lattice $\mathbb{Z}_d$ lattice of length $N$. In a normal random walk, if we let $N$ get large the end position has a probability distribution (PDF) that looks ...
-5
votes
1answer
430 views

Conics passing through integer lattice points

Find parabola that passes through exactly 3 lattice points. Find parabola that passes through exactly 4 lattice points. Prove that for any $n\geq 0$, there is a circle that passes through exactly $n$ ...
1
vote
1answer
257 views

Integral point inside a quadrilateral

Given four points in $\mathbb{R}^2$, is there an efficient method to determine if the convex hull contains an integral point $(m,n)\in\mathbb{Z}^2$? If it helps, I can assume the convex hull is a ...