# Questions tagged [geometry-of-numbers]

The geometry-of-numbers tag has no usage guidance.

80
questions

38
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0
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656
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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, ...

2
votes

0
answers

152
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### 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 ...

1
vote

1
answer

81
views

### Probability density function for the polar sine of uniformly distributed points on the sphere

If I sample three points independently, uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their polar sine?
More generally, for $k<n$ if I ...

1
vote

0
answers

52
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 ...

1
vote

0
answers

76
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 ...

4
votes

2
answers

157
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### 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 ...

2
votes

1
answer

177
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### Proof of generalized Siegel's mean value formula in geometry of numbers

Let $\mu$ be the Haar measure defined on the space of unimodular lattices, identified with $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$.
The classical Siegel's formula in geometry of numbers states ...

3
votes

1
answer

178
views

### Successive minima and the basis of lattice

I am able to prove the following two propositions: Recall that the $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-...

0
votes

0
answers

50
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### 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 $\...

6
votes

2
answers

458
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### 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,\...

3
votes

0
answers

61
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### 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 ...

4
votes

0
answers

145
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 ...

2
votes

2
answers

263
views

### Determinants of minors occurring 'within' determinant of full matrix

$A= (a_{ij})$ is an $n\times n$ symmetric positive matrix. It induces a quadratic form $f(x):= x^tAx$ on $\mathbb{R}^n$. $D_m$ denotes the determinant of the top left $m\times m$ submatrix of $A$ (or ...

1
vote

1
answer

185
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### Bounding the fractional parts of the $p^{\text{th}}$ roots of $n,n^2,...,n^{p-1}$

EDIT (August 9, 2021): I would like to ask a more general question. The original question that was fully answered is below the line.
For a positive real number $x$, denote the fractional part $x-[x]$ ...

6
votes

1
answer

143
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### 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 $\mathbb{Z}^n$, ...

9
votes

1
answer

224
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 $n$...

1
vote

1
answer

176
views

### Lattice points in hypercubes

Let $ (\Lambda_n) $ be a family of lattices, $ \Lambda_n \subset \mathbb{Z}^n $, with $ \det\Lambda_n \sim n $ as $ n \to \infty $ (meaning $ \lim_{n\to\infty} n^{-1} \det\Lambda_n = 1$). I am ...

5
votes

1
answer

162
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 ...

4
votes

1
answer

253
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 ...

0
votes

1
answer

245
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### Alternative reference to Davenport's Analytic Methods for geometry of numbers?

I was wondering if someone would be willing to suggest an alternative reference to Davenport's book Analytic Methods for Diophantine Equations and Diophantine Equations. I like the book but I would ...

6
votes

1
answer

359
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### 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 ...

7
votes

1
answer

471
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### 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\...

4
votes

2
answers

286
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
$$
w^2 - ax^2 -by^2 + abz^2 = 1
$$
$$
\...

1
vote

1
answer

155
views

### Lowering $i$th shortest vector of a lattice

LLL guarantees that we can find a basis $v_1,\dots,v_n$ of a lattice in $\mathbb R^n$ with
$$\|v_i\|\leq \gamma_{i,n} \det(\Lambda)^{1/(n-i+1)}$$ where $\gamma_i$ is a function only of $i$ and $n$.
...

2
votes

1
answer

141
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### Which lattices are rotatable into their scaled copy?

Let $L=\{\sum_i n_iv_i\mid n_i\in\mathbb Z\}$ be some lattice generated by $d$ independent vectors $(v_i)_1^d$ from $\mathbb R^d$. Call $L$ rotatable if for some $M$, a scalar multiple of some ...

9
votes

0
answers

344
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### 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 ...

15
votes

1
answer

494
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
$$\# ...

-1
votes

2
answers

228
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### On distribution of size of integer points in a subspace associated to a linear diophantine equation

Take $A,B,C,D$ pairwise coprime with $$n<A,B,C,D<2n$$ $$
n/4<|A−B|,|C−D|,|A−C|,|A−D|,|B−C|,|B−D|$$ and consider the space of solutions to $ACa+ADb+BCc+BDd=0$ spanned by $3\times 4$ matrix
$$N=...

2
votes

1
answer

308
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### Intuition behind the proof of key step in Minkowski's second inequality on successive minima

I recently knew of this note in which Prof. M. Henk presents a proof of Minkowski's second inequality on successive minima which is (purportedly) based on ideas in Minkowski's original proof. Let me ...

5
votes

0
answers

126
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 ...

0
votes

0
answers

143
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### 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 ...

1
vote

0
answers

129
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### $L^\infty$ 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 $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$
...

5
votes

0
answers

597
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### 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+...

1
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0
answers

45
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### 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 ...

11
votes

4
answers

439
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 ...

1
vote

0
answers

123
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### 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 &...

5
votes

2
answers

460
views

### Counting points on lattices in inside a box- Geometry of numebrs

Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and let $|\mathbf{x}|$ denote the L2 norm. There is a fairly standard argument involving successive minima to obtain the estimate on $N(R)$ which is the ...

2
votes

0
answers

55
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### 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 ...

8
votes

1
answer

650
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### 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 ...

1
vote

0
answers

114
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### 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 ...

6
votes

0
answers

262
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### 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 ...

2
votes

3
answers

321
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### Geometry of numbers argument: counting integers with some linear condition

I am interested in the proof of the following result:
Suppose that $A > 1$, $\lambda \in \mathbb{R}$, and for $0 < Z \leq 1$, let $U(Z)$ be the number of integer solutions $v$ of
\begin{...

6
votes

0
answers

113
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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 ...

1
vote

1
answer

82
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### Dual lattices up to a q scaling factor

In this paper : https://eprint.iacr.org/2011/501.pdf
There is an equality page 10, in the second paragraph considered by the authors as "easy to check". If someone could explain to me why the set at ...

4
votes

2
answers

5k
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### 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:
$n = a^2 + b^2 + c^2$ ...

6
votes

1
answer

186
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 $\...

3
votes

2
answers

2k
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### On successive minima and basis of a lattice

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 ...

6
votes

1
answer

625
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### 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 ...

3
votes

0
answers

188
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### 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, ...

5
votes

1
answer

325
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, ...