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42 views

Number of lattice points inside a parallelogram defined by a vector on the “rhombic plane” [on hold]

I will give here a specific case and try to explain the question the best I can: The rhombic plane is the plane with the x-axis the same as the Cartesian plane but the y-axis tilted at $60$ degrees. ...
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0answers
130 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
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0answers
74 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 ...
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0answers
86 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_{ ...
2
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0answers
83 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} ...
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1answer
207 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 ...
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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 ...
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1answer
355 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 ...
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3answers
542 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
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0answers
93 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 ...
2
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0answers
170 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 ...
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2answers
387 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 ...
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2answers
397 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 ...
5
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0answers
243 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 ...
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0answers
742 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
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1answer
566 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, ...
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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 ...
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0answers
118 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 ...
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1answer
444 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
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1answer
608 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 ...
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2answers
548 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
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403 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
308 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
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1answer
508 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 ...
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2answers
652 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
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1answer
481 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
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2answers
543 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 ...