The tag has no wiki summary.

learn more… | top users | synonyms

5
votes
1answer
142 views

Maximum sets of lattice points such that only a few points collinear

Consider all the integer points $\in [0,n]\times[0,n]$, I want to find the maximum subset $S$ of which such that there are at most $n^\varepsilon(0<\varepsilon<1)$ points in $S$ collinear. So, ...
6
votes
0answers
119 views

Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying $b$ is bilinear, ...
6
votes
1answer
78 views

space of reduced positive definite quadratic forms

What is the highest dimension for which the space of reduced positive definite quadratic forms (or the fundamental domain of $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$) has been explicitly calculated? I know ...
1
vote
0answers
171 views

Average rank of elliptic curves over function fields

De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the ...
2
votes
0answers
81 views

The number of different lattice triangles

Two convex lattice polygons are equivalent if there is a lattice-preserving affine transformation mapping one of them to the other. Equivalent polygons have the same area. Let $H(A)$ denote the number ...
1
vote
0answers
105 views

Siegel's Mean Value Theorem by Rogers and Macbeath

I recently became engaged in the work of Siegel, Schmidt, Rogers, Macbeath regarding random lattices and geometry of numbers, e.g. Siegel proved that $$\int_{SL(n,\mathbb{R})/SL(n,\mathbb{Z})} \sum_{ ...
3
votes
0answers
88 views

On one class of Euclidean lattices

Let $\Lambda\subset \mathbb Z^3$ be 3D lattice with a basis $$a_1=\left(\begin{smallmatrix} a_{11} \\ a_{21}\\ a_{31} \end{smallmatrix}\right),a_2=\left(\begin{smallmatrix} a_{12} \\ a_{22}\\ a_{32} ...
5
votes
1answer
225 views

Regular lattice polygons

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within ...
3
votes
0answers
121 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 ...
18
votes
2answers
1k views

Are most curves over Q pointless?

Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : Most ...
15
votes
1answer
385 views

Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem: What is the volume of the largest symmetric convex subset ...
6
votes
3answers
555 views

Sums of inverse determinants over matrices

Let $A \in M_n(\mathbb Z)$ and $\|A\| = \max |a_{ij}|$. Denote $$ S(r) = \sum_{\substack{\|A\| \leq r \\\ \det{A} \neq 0}} \dfrac{1}{|\det{A}|} $$ - the sum over all matrices $A \in M_n(\mathbb Z)$ ...
7
votes
0answers
98 views

Minimal number of colours for colouring Voronoi-cells of a $d-$dimensional lattice

There are arbitrarily large finite sets of points in $\mathbb R^3$ whose Voronoi-domains intersect all pairwise in $2-$dimensional polytopes. This shows that one needs infinitely many colours in order ...
1
vote
0answers
197 views

Gauss circle problem and Jacobi-type estimates for higher dimensions

Hello everyone, I was doing some late night random reading and I got to wonder about some stuff about the Gauss circle problem. To begin with, consider a circle in $\mathbb{R}^{2}$ with centre at the ...
5
votes
2answers
394 views

On Weil's characters of type (A)

In Weil's paper "On a certain type of characters of the idele-class group of an algebraic number field", Weil introduces a class of characters on the Idele class group (of not necessarily finite ...
5
votes
2answers
405 views

Still more generalized Dirichlet Theorem

Dirichlet proved a classical theorem about approximating irrational real numbers with rational numbers, saying that for any irrational real number $\alpha$, you can find infinitely many rational ...
9
votes
0answers
288 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...
29
votes
0answers
855 views

(Approximately) bijective proof of $\zeta(2)=\pi^2/6$ ?

Given $A,B\in {\Bbb Z}^2$, write $A \leftrightarrows B$ if the interior of the line segment AB misses ${\Bbb Z}^2$. For $r>0$, define $S_r:=\{ \{A, B\} | A,B\in {\Bbb Z}^2,||A||<r,||B||<r, ...
14
votes
1answer
576 views

Totally rational polytopes

Define a convex polytope in $\mathbb{R}^d$ as totally rational (my terminology) if its vertex coordinates are rational, its edge lengths are rational, its two-dimensional face areas are rational, ...
10
votes
3answers
1k views

Number of lattice points in a random disk of radius r

Consider a disk of radius $r$ centered at $(x,y)$, where $(x,y)$ is chosen from the uniform distribution on $[0,1) \times [0,1)$, and let the random variable $N$ be the number of lattice points in the ...
1
vote
0answers
119 views

A bounded function of the packing and covering density of lattices

Given a (finite-dimensional) lattice $L$ of an Euclidean vector-space, the function $$L\longmapsto -\log(\hbox{packing density of }L)/ \log(\hbox{covering density of }L)$$ is bounded and bounded away ...
20
votes
1answer
469 views

Voronoi cell of lattices with the same profile

Definition 1. Given a body $V$ in $\mathbb R^n$, the function $p_V\colon \mathbb R_+\to \mathbb R_+$ $$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$ will be called profile of $V$. Definition 2. Define ...
8
votes
1answer
611 views

Projecting the unit cube onto a (very special) subspace

Let $n>1$ be an integer, and $a>1$ a real number. Consider the subspace $L<R^{2^n}$ generated by the $n$ possible tensor products of the $n-1$ copies of the vector $(1,a)$ and one copy of ...
2
votes
2answers
550 views

Projecting the unit cube onto a subspace [closed]

I have some (rather exotic) subspace $L<R^n$, and I want to show that every non-zero vector in $\{0,1\}^n$ has a relatively small projection onto $L$. What general results and tools can be helpful? ...
7
votes
0answers
428 views

Higher-dimensional analogs of the Farey sequence/Riemann hypothesis connection?

See here for Franel and Landau's equivalent forms of the Riemann hypothesis in terms of the uniformity of distribution of Farey sequences. ...
4
votes
0answers
314 views

An operator-norm version of Siegel's Lemma

Is there a kind of Siegel's Lemma saying that if $M$ is a ``small-height'' integer matrix, then there is a "small-height" vector $x$ with $\|Mx\|=\|M\|\|x\|$? (Here $\|Mx\|$ and $\|x\|$ denote the ...
17
votes
1answer
533 views

Is the norm of a $0-1$ matrix (almost) attained on a $0-1$ vector?

I'd like to state explicitly a problem which was somehow hidden in my three-week-old post: Does there exist an absolute constant $c>0$ with the property that for any matrix $M\in{\mathcal ...
6
votes
2answers
672 views

Is the operator norm always attained on a $\{0,1\}$-vector?

Given an operator $f\colon R^m\to R^n$, can one always find a non-zero vector $x\in \{ 0,1 \}^m$ such that $\|f(x)\|/\|x\|\ge0.01\|f\|$? (Here I denote by $\|\cdot\|$ both the Euclidean norms in $R^m$ ...
7
votes
1answer
486 views

Verifying my other example in the Geometry of Numbers and Quadratic Forms

In answer to Pete L. Clark's question Must a ring which admits a Euclidean quadratic form be Euclidean? on Euclidean quadratic forms, I also gave an example in six or fewer variables, repeated below. ...
7
votes
2answers
550 views

Verifying an example in the Geometry of Numbers and Quadratic Forms

In answer to Pete L. Clark's question Must a ring which admits a Euclidean quadratic form be Euclidean? on Euclidean quadratic forms, I gave an example in seven variables, repeated below. Pete's ...