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 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 $a$ ...
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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 ...
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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 ...
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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 $A \in \mathbb Z^{n \times m}_q$ ...
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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 ...
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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?
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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194 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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640 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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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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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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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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101 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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66 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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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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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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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 ...