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. […]

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 […]

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 […]

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& […]

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)?

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 […]

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 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 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 […]

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 […]

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