Questions tagged [geometry-of-numbers]

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Can a regular icosahedron contain a rational point on each face?

The title says it all: Is there a (regular) icosahedron containing a rational point on each of its faces? For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, ...
Ilya Bogdanov's user avatar
9 votes
0 answers
378 views

Number fields ordered by discriminant

Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are ...
Vesselin Dimitrov's user avatar
9 votes
0 answers
458 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 ...
Jyrki Lahtonen's user avatar
8 votes
0 answers
525 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. http://en.wikipedia.org/wiki/Farey_sequence#...
David Feldman's user avatar
7 votes
1 answer
499 views

Counting points on the intersection of a box and a lattice

Let $A:\mathbb{Z}^n\to \mathbb{Z}^n$ be non-singular. Consider a box $B=[0,N_1]\times [0,N_2] \times \dotsc \times [0,N_n]$. Let $p_1,\dotsc,p_n$ be primes (distinct, if you wish) and let $L = p_1\...
H A Helfgott's user avatar
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7 votes
0 answers
190 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, $b(x,...
eins6180's user avatar
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7 votes
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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 ...
Roland Bacher's user avatar
6 votes
0 answers
266 views

Bound on the determinant of a quadratic form restricted to a subspace

Let $Q\colon \mathbb{Z}^{n}\oplus\mathbb{Z}^m\to\mathbb{R}$ be a real quadratic form, which we denote $Q(x,y)$, $x\in\mathbb{Z}^n$, $y\in\mathbb{Z}^m$. Suppose: The minimum of $Q(x,y)$ as $y$ varies ...
Yoav Kallus's user avatar
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6 votes
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Rational $d$-simplices

Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$ such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational. So a rational triangle has rational edge lengths and ...
Joseph O'Rourke's user avatar
5 votes
0 answers
128 views

Averaging number of lattice points in a box over a family of lattices

Consider the diophantine equation $$ x_1y_1^3 + \dots + x_s y_s^3 = 0. $$ For fixed $\mathbf{y}$ with coprime coordinates this is a $s-1$ dimensional lattice $\Lambda(\mathbf{y})$. Let $N(X)$ denote ...
leithian's user avatar
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0 answers
602 views

Necessary and Sufficient condition for Sharpness of Bombieri and Vaaler's result on Siegel's lemma?

This Wikipedia page currently quotes Bombieri and Vaaler's result on Siegel's lemma: Suppose we are given a system of $m$ linear equations in $n$ unknowns such that $n>m$, say $a_{11}x_1+\dots+...
Turbo's user avatar
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5 votes
0 answers
390 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 ...
Seva's user avatar
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4 votes
0 answers
151 views

Closest integer point to a sphere with radius R

I have a sphere in $\mathbb{R}^d$ with radius $R$ whose center is not necessarily the origin. I am interested in the closest integer lattice point to it. Indeed, it depends on the center location, but ...
Morteza's user avatar
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3 votes
0 answers
64 views

Stability of successive minima with respect to the metric on the space of lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
No One's user avatar
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3 votes
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192 views

Counting number of points in a lattice with bounded length

I am interested in counting number of lattices using the following theorem. The following is Theorem IV (page 412) in Chapter VIII of "An introduction to the geometry of numbers (second printing, ...
SJY's user avatar
  • 579
2 votes
0 answers
170 views

Mistake in Rogers' paper: "number of lattice points in a set" for the case $n=2$?

Let $f:\mathbb R^n\to \mathbb R$ be a nonnegative Borel measurable function, and let $f^*$ be the function obtained from $f$ by spherical symmetrization (see Rogers' paper: number of lattice points in ...
taylor's user avatar
  • 425
2 votes
0 answers
58 views

Configurations of minimal vectors for a 4-dimensional symplectic lattice

The possible configurations of minimal vectors for a 4-dimensional lattice are known for ages, but what about symplectic lattices ? If a 4-dimensional symplectic lattice $\Lambda$ has two minimal ...
crispus's user avatar
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2 votes
0 answers
415 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 ...
Pig's user avatar
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1 vote
0 answers
54 views

Second moment version of the multiple-sum Rogers integration formula

I know the following theorems due to Rogers. Let $X$ denote the space of $n$-dimensional unimodualar lattices in $\mathbb R^n$, equipped with the canonical Haar measure. Theorem 1(Siegel-Rogers). Let ...
taylor's user avatar
  • 425
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 ...
user95246's user avatar
  • 237
1 vote
0 answers
129 views

$L^\infty$ norm lower bound for Integer points in null spaces of recursively defined integer vectors?

Letting $\otimes$ be matrix kronecker/tensor product with $n\in\Bbb N$ as a parameter define non-negative integer vectors recursively $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$ ...
Turbo's user avatar
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1 vote
0 answers
45 views

Integral matrices with a lot of small integral vectors in the kernel

Suppose $A$ is an $m\times n$ matrix with integer coefficients. These coefficients are possibly very large, however we assume there are at least $K_C$ vectors $x\in\mathbb Z^n$ with $\max_i |x_i|\leq ...
Brandon Hanson's user avatar
1 vote
0 answers
124 views

Computing the successive minima of the following lattice in $\mathbb{R}^4$

Let us define the lattice $\Lambda$ in $\mathbb{R}^4$ defined by the matrix $$ \Lambda = \begin{bmatrix} A & 0 & 0 & 0 \\ 0 & A & 0 & 0 \\ \gamma_1 & \gamma_2 &...
Johnny T.'s user avatar
  • 3,547
1 vote
0 answers
114 views

On the number of $\mathbb Z$-linearly independent integer points in a bounded region

Let $\|\,\,\,\|$ denote the Euclidean norm on $\mathbb R^2$. Let $\Lambda$ be a sublattice of $\mathbb Z^2$ and $m < M$ be positive real numbers. We say that a point $(x,y)$ in $\mathbb Z^2$ is ...
Anton's user avatar
  • 1,573
1 vote
0 answers
481 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 ...
Cosmin Pohoata's user avatar
1 vote
0 answers
135 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 ...
Roland Bacher's user avatar
0 votes
0 answers
66 views

Extension of primitive set of vectors and reduction theory

Let $\Lambda$ be a unimodular lattice in $\mathbb R^d$ (unimodularity is not really necessary here but just for convenience) and let $B$ be a ball centered at the origin that contains $(k+1)$-many $\...
taylor's user avatar
  • 425
0 votes
0 answers
144 views

Siegel's Mean Value Theorem by Macbeath and Rogers

It is claimed in an answer in mathoverflow to a question about Siegel's Mean value theorem (link- Siegel's Mean Value Theorem by Rogers and Macbeath) that there is mistake for the case $n=2$. I ...
mahbubweb's user avatar
  • 111