The geometry-of-numbers tag has no wiki summary.

**34**

votes

**1**answer

1k 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, ...

**9**

votes

**0**answers

317 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 ...

**7**

votes

**0**answers

102 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 ...

**7**

votes

**0**answers

445 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.
...

**6**

votes

**0**answers

134 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,
...

**5**

votes

**0**answers

329 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 ...

**3**

votes

**0**answers

131 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, ...

**3**

votes

**0**answers

93 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}
...

**2**

votes

**0**answers

93 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

**0**answers

212 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 ...

**1**

vote

**0**answers

143 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_{ ...

**1**

vote

**0**answers

241 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 ...

**1**

vote

**0**answers

120 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 ...