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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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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Lattice Paths from $(1, 1) \to (x, y)$ [on hold]

Moderator Note: This is a current contest question on Brilliant.org. 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 ...
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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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65 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 ...
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167 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 ...
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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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207 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, ...
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187 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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625 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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58 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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87 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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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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99 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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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 ...
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45 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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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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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 $P$ ...
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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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220 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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104 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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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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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 ...
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120 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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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 ...
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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 ...
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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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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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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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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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131 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 ...