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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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 ...
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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, ...
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1answer
40 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 ...
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2answers
48 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 ...
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25 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 ...
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1answer
33 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 ...
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1answer
122 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 ...
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2answers
53 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$ ...
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0answers
8 views
Find the centroid of an inequality-bounded integer lattice.
Let $Ax \ge c$ be a system of $k$ linear inequalities that define a bounded region in $\mathbb{R}^n$. Suppose we assign each point on the integer lattice (i.e. all coordinates are integers) of ...
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5answers
59 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?
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48 views
A set of lattice points
How can one define a set of lattice points in a plane ?
I searched for it, but I did not found a precise definiton.
Thank you for helping
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29 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 ...
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0answers
17 views
Index of the sum of two lattices determined by rational planes.
I am wondering if the following is true:
Let us for simplicity's sake consider $\mathbb{Z}^3 \subset \mathbb{R}^3$,
and let $v,w \in \mathbb{Z}^3$ be primitive, linearly independent vectors.
Let $L$ ...
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1answer
44 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 ...
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1answer
17 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 ...
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25 views
Property lattice - sign of variables
I am doing program analysis and am interested in the sign of variables. That is, whether we can guarantee that for a given program point and a variable x (at least) one of the following properties ...
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1answer
25 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 ...
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20 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 ...
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1answer
19 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$ ...
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1answer
113 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 ...
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2answers
107 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 ...
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49 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}$. ...
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1answer
39 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 ...
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1answer
80 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:
...
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1answer
97 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 ...
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47 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.
...
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1answer
31 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 ...
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1answer
111 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 ...
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2answers
93 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 ...
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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 ...
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1answer
67 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. ...
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1answer
85 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 ...
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1answer
52 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 ...
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1answer
73 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?
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0answers
32 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 $\nu$ ...
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2answers
63 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 ...
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0answers
36 views
Lattice Reduction Problem: Minimizing the Longest 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 ...
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1answer
55 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 ...
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1answer
114 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 ...
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0answers
68 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 ...
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1answer
65 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, ...
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1answer
79 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 $n$ ...
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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 ...
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2answers
128 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 ...
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1answer
113 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 ...
2
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1answer
111 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 ...
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1answer
36 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 ...
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59 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 ...
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1answer
148 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 ...
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2answers
261 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 ...
