Articles of integer lattices

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 S is closed, then it suffices to take $Volume(S) \geq 2^ndet(L)$.” I am wondering if the condition for boundedness can be relaxed. […]

Measure theoretic proof of $|\Bbb{Z}^d/A\Bbb{Z}^d| = |\det(A)|$

Let $A \in \Bbb{Z}^{d\times d}$ be an invertible matrix with entries in $\Bbb{Z}$. It is well-known (and can be proved using algebraic properties of matrices) that the index of the group $A \Bbb{Z}^d \leq \Bbb{Z}^d$ in $\Bbb{Z}^d$ is given by $$|\Bbb{Z}^d/A\Bbb{Z}^d| = |\det(A)|. \qquad (\dagger)$$ Looking at the right hand side of this question, I […]

Lattice generated by vectors orthogonal to an integer vector

Given a non-zero vector $\boldsymbol{v}$ composed of integers, imagine the set of all non-zero integer vectors $\boldsymbol{u}$, such that $\boldsymbol{u} \cdot \boldsymbol{v} = 0$, i.e., the integer vectors orthogonal the original vector. The set $S = \{\boldsymbol{u} : \boldsymbol{u} \cdot \boldsymbol{v} = 0\}$ seems to form a $dim(\boldsymbol{v})-1$ dimensional lattice. Specifically, it’s clear that for […]

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=\begin{pmatrix} 0& 0& 1\\ 1& 0& 0\\ 0& 1& 0 \end{pmatrix}, \text{ and }\;\; A_2=\begin{pmatrix} 0& 1& 0\\ -1& 0& 0\\ 0& 0& […]

Count the number of elements of ring

1/ How to count the number of elements of $\mathbb{Z}[i]/(1+2i)^n$? 2/ How to write $\mathbb{Z}[i]/(1+2i)^n$ as direct sum of cyclic groups (in view of the structure theorem of finite abelian groups)?

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’m especially interested in the case $m=3$. For $m=2$, we have the following: If $a$ and $b$ are (positive) relatively prime integers, then you can always find integers $c$ and […]

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 is something someone has studied so hopefully someone here could point me to some references. Also, consider the lattice points […]

Ballot counting when ties occurs exactly $r$ times

Ballot Problem with Fixed Number of Ties: Problem Statement: In an election, candidate A receives $m$ votes and candidate B receives $n$ notes. Let $m \ge n$. In how many ways can the ballots be counted so that ties occur exactly $r$ times ($r \le n$)? Example: $m = 3, n = 2, r = […]

Pick's Theorem on a triangular (or hex) grid

Pick’s theorem says that given a square grid consisting of all points in the plane with integer coordinates, and a polygon without holes and non selt-intersecting whose vertices are grid points, its area is given by: $$i + \frac{b}{2} – 1$$ where $i$ is the number of interior lattice points and $b$ is the number […]

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 $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains (draw their lattices in the complex plane). In my class we drew a general lattice for $\mathbb{Z}[i]$: and […]