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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How can I find the number of the shortest paths between two points on a 2D lattice grid?

How do you find the number of the shortest distances between two points on a grid where you can only move one unit up, down, left, or right? Is there a formula for this? Eg. The shortest path between ...
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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 $\vec x$...
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Elementary proof that if $A$ is a matrix map from $\mathbb{Z}^m$ to $\mathbb Z^n$, then the map is surjective iff the gcd of maximal minors is $1$

I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by an $m \times n$ matrix $A$, then the map is surjective iff the gcd of ...
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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 20×...
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Which internal angles can a lattice polygon have?

I am wondering if for a lattice polygon an internal angle can take any value? If no which ones not and why? I guess there will be some restrictions due to the discrete nature of the grid but I am ...
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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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Ideals of $\mathbb{Z}[i]$ geometrically

It is pretty easy to visualize the ideals of $\mathbb{Z}$ in the "integer line". Let's go up to $\mathbb{Z}[i]$ and consider the ideal $3\cdot\mathbb{Z}[i]$. We can visualize it as a "sub-lattice" ...
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$x+y\sqrt{2}$ infimum ($x,y\in \mathbb{Z}$)

I've looked for help with this question but I have not found anything, I hope this is not a duplicate. Define the set $A=\{\mid x+y\sqrt{2}\mid \ : x,y\in \mathbb{Z}\ \mbox{and} \mid x+y\sqrt{2}\mid\...
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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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Prove that $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains

For a non-square integer $d$ such that $d \equiv 1 \mod 4,$ we define the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}.$$ Prove that $\...
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Integer lattice points on a sphere

Suppose we have a sphere centered at the origin of $\mathbb{R^{n}}$ with radius $r$. Are there known theorems that state the number of integer lattice points that lie on the sphere? It seems like this ...
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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 ...
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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? I'...
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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 \...
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Expression from generators of Special Linear Groups II

I wonder whether one can generate this $t$ matrix form the $A_1$ and $A_2$ matrix below. Here $$ t=\begin{pmatrix} 1& 1& 0\\ 0& 1& 0\\ 0& 0& 1 \end{pmatrix} $$ from: $$ A_1=\...
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Proving this identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$ using lattice paths

How can I prove the identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$? I have to prove it using lattice paths, it should be related to Catalan numbers The $n$th ...
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The $n$-shortest lattice vectors problem in $\mathcal{R}^2$

I am looking for an algorithm to compute the $n$ shortest lattice vectors in $\mathcal{R}^2$. The problem statement is as follows: Given a lattice $L: \{ m \vec{u}+n\vec{v} \} \in \mathcal{R}^2$, a ...
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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 ...
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Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
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Projection of a lattice onto a subspace

Let $G$ be a $n \times n$ matrix with real entries and let $\Lambda = \{x^n \colon \exists i^n \in \mathbb{Z}^n \text{ such that } x^n = G \cdot i^n\}$ define a lattice. I am interested in projecting ...
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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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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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How to find the number of squares formed by given lattice points?

Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points? Additionally, if we were to ...
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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 and ...
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A book useful to learn lattices (discrete groups)

Does anyone know a good book about lattices (as subgroups of a vector space $V$)?
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Naive Grouping for Factorization

I have a naive grouping method for factorization. I am curious as to its novelty and aspects of the code below that will increase its efficiency. The method is best described with an example: For n ...
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Lattice Paths to $(2n, 2n)$ that do not go through $(n,n)$

I know that from the origin to $(x,y)$ there are $${x+y \choose x} = {x+y \choose y}$$ Lattice paths. The question is finding the number of paths from the origin to $(2n, 2n)$ not passing through $(n,...
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Minimal diameter of a connected subset of $\mathbb{Z}^d$

Let $A \subset \mathbb{Z}^d$ be a connected set of cardinality $|A|$. Let the diameter of the $A$ be defined as the length of the longest path of distinct, nearest neighbor sites which is entirely ...
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Newton-Raphson For Integer Factorization

Per my earlier question on Naive Grouping for factorization here, below is the modified Newton-Raphson method (integers only) for the polynomial $N -x^2 - yx - x = 0$. ...
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topology on n-colorings of a lattice?

Consider the collection of all $n$-colorings of $\mathbb{Z^{d}}$ (i.e. the collection of all ways to color each lattice point one of $n$ colors). What are some non-trivial ways to define a topology on ...
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Lattice Paths Question

You know, how we can have lattice paths, where we can move either one block north, or one block east, and we have the find all the possible ways of reaching the point (x.y) from (0,0). That is $\...