All Questions
14 questions
12
votes
2
answers
5k
views
The Gauss circle problem on a hexagonal lattice
Take an infinite hexagonal lattice (or equivalently, an equilateral triangular lattice), with unit spacing between the closest lattice point pairs, and draw a disc of radius $r$ centered on a lattice ...
- 197
7
votes
2
answers
453
views
Bound on Minimal Length of Vectors in Lattice and its Dual Lattice
Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
- 679
5
votes
1
answer
190
views
Finding a superbase in a lattice of Voronoi first kind
An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that
$\{b_1,\ldots,b_n \}$ is a ...
user14875
4
votes
1
answer
108
views
Closed cobounded additive submonoid of $\mathbb{R}^n$
Let $M$ be a closed additive submonoid of $\mathbb{R}^n$ with $n\geq1$. Suppose also that there exists $r>0$ such that every ball of radius $r$ intersects $M$. I wonder if we can obtain more ...
- 143
4
votes
1
answer
293
views
Number of points in a lattice and an oblong box
I have a very simple question in geometry of numbers. (It is a slight modification of Counting points on the intersection of a box and a lattice .) There's a bound I can easily prove, and it's good ...
- 20.2k
4
votes
0
answers
111
views
Advice on results for balls on regular $N$-dimensional grids
I have obtained some results regarding balls on regular $N$-dimensional grids. I would like expert opinion on wether the results are significant or interesting enough for (trying to) publish them in a ...
- 337
3
votes
2
answers
202
views
Existence of lattices whose circles have bounded number of points
For any plane lattice $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, with $A,B$ linearly independent vectors in $\mathbb R^2$, we define the set of the circles in $\Lambda$ as
$$\mathcal K(\Lambda) = \...
- 315
3
votes
1
answer
518
views
n-dimensional Delaunay Triangulation of Lattices
I have several questions concerning the Delaunay triangulation of a high dimensional lattice.
Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
- 31
3
votes
1
answer
381
views
Source on counting lattice points on a line
Looking for a book or article on the result linked below. The result tells us that the number of lattice points on a line between points $(a,b)$ and $(c,d)$ is given by $\gcd(a-c,b-d)+1$.
https://math....
- 45
2
votes
0
answers
95
views
Is there an exact solution for the number of points within a circle of radius r for an honeycomb lattice?
I want to ask if exists an exact solution for the number of points within a circle of radius r for an honeycomb lattice.
I know that it is exist for an square lattice https://mathworld.wolfram.com/...
- 31
1
vote
1
answer
228
views
Lattice points in hypercubes
Let $ (\Lambda_n) $ be a family of lattices, $ \Lambda_n \subset \mathbb{Z}^n $, with $ \det\Lambda_n \sim n $ as $ n \to \infty $ (meaning $ \lim_{n\to\infty} n^{-1} \det\Lambda_n = 1$). I am ...
- 503
1
vote
0
answers
77
views
Lattice packing
Let $\Lambda$ be a lattice in $R^n$ and $R>0$ a real number.
Consider the number $N$ of points in $\Lambda$ of norm less than $R$. Let $R$ goes to infinity. What can be said about the asymptotic ...
- 237
1
vote
0
answers
124
views
Number of lattice points in a structural symmetric convex body
Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space
\begin{equation}
\...
- 550
1
vote
0
answers
60
views
Finding special vectors generated by a matrix
Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix.
Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...
- 13.9k