Questions tagged [geometry-of-numbers]

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(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\} \mid A,B\in {\Bbb Z}^2,\|A\|<r,\|B\|<...
David Feldman's user avatar
43 votes
0 answers
1k views

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
38 votes
3 answers
2k views

Is there a regular pentagon with a rational point on each edge?

This question was asked by Yaakov Baruch in the comments to the question Can a regular icosahedron contain a rational point on each face? It seems that this question deserves special attention.
Alexey Ustinov's user avatar
26 votes
2 answers
2k 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 ...
Chandan Singh Dalawat's user avatar
22 votes
1 answer
646 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 ...
Anton Petrunin's user avatar
19 votes
1 answer
887 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 M}_{m\...
Seva's user avatar
  • 22.8k
17 votes
1 answer
1k 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, etc.,...
Joseph O'Rourke's user avatar
15 votes
1 answer
530 views

Counting primitive lattice points

In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result): Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then $$\# ...
Daniel Loughran's user avatar
15 votes
1 answer
595 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 $S$...
zeb's user avatar
  • 8,533
11 votes
4 answers
446 views

Sequential addition of points on a circle, optimizing asymptotic packing radius

Suppose I have to put $N$ points $x_1, x_2, \ldots, x_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced ...
Yoav Kallus's user avatar
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11 votes
3 answers
691 views

Counting points on lattices

I expect that the following is a standard problem from analytic number theory, but I don't know where exactly to look for an answer. Let f: ℤr→ H be a surjective homomorphism into a ...
Tzanko Matev's user avatar
11 votes
3 answers
2k 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 ...
James Propp's user avatar
  • 19.4k
9 votes
1 answer
227 views

Are the class numbers of $\mathbb{Q}(\cos(2\pi / m))$ $O(m^n)$ for some fixed $n$?

Question: Are the class numbers of $\mathbb{Q}(\cos(\frac{2\pi}m))$ $O(m^n)$ for some fixed $n$? Evidences (e.g. a recent paper) showing that the question above is open are also OK. Remark: If such $n$...
LeechLattice's user avatar
  • 9,421
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
1 answer
1k 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_{ ...
Soeren's user avatar
  • 101
8 votes
1 answer
727 views

Minkowski's Linear Forms Theorem With Complex Coefficients

Minkowski's Linear Forms Theorem is often stated about linear forms with real coefficients. However, in Narkiewicz's Elementary and Analytic Theory of Algebraic Numbers, the following generalization ...
Greg K's user avatar
  • 201
8 votes
1 answer
645 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 $(a,...
Seva's user avatar
  • 22.8k
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
2 answers
638 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 ...
Will Jagy's user avatar
  • 25.3k
7 votes
1 answer
547 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. ...
Will Jagy's user avatar
  • 25.3k
7 votes
2 answers
283 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 ...
seungki's user avatar
  • 91
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
  • 19.3k
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
  • 1,302
7 votes
0 answers
128 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 ...
Roland Bacher's user avatar
6 votes
3 answers
647 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 $A \in M_n(\mathbb Z)$ ...
Anton Menshov's user avatar
6 votes
1 answer
382 views

The number of quadratic forms attaining Hermite's constant

$\require{AMScd}$ I'm considering minimum values (at non-zero integer points) of real, positive-definite, quadratic forms of determinant $1$. These are functions $f:\mathbb{R}^n\to \mathbb{R}$ which ...
Chertopkhanov on Malek Adel's user avatar
6 votes
2 answers
559 views

Counting points on lattices in inside a box- Geometry of numbers

Let $\Lambda$ be a lattice in $\mathbb{R}^n$, and let $|\mathbf{x}|$ denote the $L^2$ norm. There is a fairly standard argument involving successive minima to obtain an estimate on $N(R)$ which is the ...
Johnny T.'s user avatar
  • 3,547
6 votes
1 answer
651 views

Counting number of points in a lattice with bounded sup norm

Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let $\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$ be a ...
SJY's user avatar
  • 579
6 votes
2 answers
502 views

Bounds on Bézout coefficients

Let $0<a_1 \le a_2 \le \cdots \le a_n$ be positive integers such that $a_1 + \cdots + a_n = m$ and $\gcd(a_1,\ldots,a_n)=1$. Let $\mathbf a :=(a_1,\ldots,a_n)\in\mathbb Z^n$ and $\mathbf x:=(x_1,\...
Pranay Gorantla's user avatar
6 votes
1 answer
162 views

Maximal sublattice index in Minkowski's Second Theorem

Let $B$ be a (small) convex compact set in $\mathbb{R}^n$, symmetric around the origin. Let $\Gamma$ be a lattice in $\mathbb{R}^n$ of dimension $n$ (I'm almost sure we can just take $\mathbb{Z}^n$, ...
Jakub Kamiński's user avatar
6 votes
2 answers
752 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 ...
David McKinnon's user avatar
6 votes
1 answer
195 views

Lattice parallelogram of minimal area containing convex lattice polygon

What is the minimal constant $\alpha$ so that for any convex lattice polygon $F$ there exists a lattice parallelogram $P\supseteq F$ of area $A(P)\leq \alpha\cdot A(F)$? It is not hard to show that $\...
Fedor Petrov'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
  • 5,926
6 votes
0 answers
113 views

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
2 answers
1k 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 $R^m$ ...
Seva's user avatar
  • 22.8k
5 votes
2 answers
780 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 ...
Hugo Chapdelaine's user avatar
5 votes
1 answer
559 views

Reverse Gauss's Circle Problem

Gauss's Circle Problem [1] is to find the number of lattice points inside the boundary of a circle with a given radius and center at the origin. I'm interested in the "reversed" version of this ...
Armin Meisterhirn's user avatar
5 votes
1 answer
173 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 ...
user avatar
5 votes
1 answer
512 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 ...
Igor Rivin's user avatar
  • 95.5k
5 votes
1 answer
328 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, ...
Paul's user avatar
  • 509
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
  • 163
5 votes
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
  • 13.7k
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
  • 22.8k
4 votes
2 answers
5k views

Proving Legendre's Sum of 3 Squares Theorem via Geometry of Numbers

Every positive integer can be written as the sum of 4 squares $n = a_1^2 + a_2^2 + a_3^2 + a_4^2$ however, if we only allow sum of 3 squares some numbers have to be left out: $n = a^2 + b^2 + c^2$ ...
john mangual's user avatar
  • 22.6k
4 votes
1 answer
414 views

Counting number of points on a lattice in a hypercube

Suppose I have a lattice $\Lambda \in \mathbb{R}^n$. Let $X_i >0$ for $i=1,..,n$. I am interested in some references regarding counting number of points of $\Lambda$ inside $[-X_1, X_1] \times \...
Johnny T.'s user avatar
  • 3,547
4 votes
2 answers
174 views

How large is the set of unimodular lattices whose sucesssive minima cannot be attained by a basis of lattice?

Recall that the $i$-th successive minimum of $L\in \mathcal L$ (space of full rank lattices in $\mathbb R^d$), denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-linearly ...
taylor's user avatar
  • 425
4 votes
2 answers
291 views

Quadratic diophantine equations and geometry of numbers

Let (for concreteness) $a = 2$, $b = \sqrt{5}$ and $\varphi = (\sqrt{5}+1)/2$. I am interested in solutions $(w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system $$ w^2 - ax^2 -by^2 + abz^2 = 1 $$ $$ \...
Stefan Witzel's user avatar
4 votes
1 answer
263 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 ...
H A Helfgott's user avatar
  • 19.3k
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
  • 628