Questions tagged [geometry-of-numbers]
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82
questions
38
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3
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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.
43
votes
0
answers
1k
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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
170
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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
89
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
54
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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
77
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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
173
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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
197
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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
237
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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
66
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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
498
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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
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 ...
4
votes
0
answers
151
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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
282
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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
162
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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
227
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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
197
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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
173
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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
263
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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
260
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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
381
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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
499
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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
290
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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
$$
w^2 - ax^2 -by^2 + abz^2 = 1
$$
$$
\...
1
vote
1
answer
164
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Lowering $i$th shortest vector of a lattice
LLL guarantees that we can find a basis $v_1,\dots,v_n$ of a lattice in $\mathbb R^n$ with
$$\|v_i\|\leq \gamma_{i,n} \det(\Lambda)^{1/(n-i+1)}$$ where $\gamma_i$ is a function only of $i$ and $n$.
...
2
votes
1
answer
146
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Which lattices are rotatable into their scaled copy?
Let $L=\{\sum_i n_iv_i\mid n_i\in\mathbb Z\}$ be some lattice generated by $d$ independent vectors $(v_i)_1^d$ from $\mathbb R^d$. Call $L$ rotatable if for some $M$, a scalar multiple of some ...
9
votes
0
answers
378
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Number fields ordered by discriminant
Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are ...
15
votes
1
answer
529
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Counting primitive lattice points
In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result):
Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then
$$\# ...
-1
votes
2
answers
229
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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
$$N=...
2
votes
1
answer
318
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Intuition behind the proof of key step in Minkowski's second inequality on successive minima
I recently knew of this note in which Prof. M. Henk presents a proof of Minkowski's second inequality on successive minima which is (purportedly) based on ideas in Minkowski's original proof. Let me ...
5
votes
0
answers
128
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Averaging number of lattice points in a box over a family of lattices
Consider the diophantine equation
$$ x_1y_1^3 + \dots + x_s y_s^3 = 0. $$
For fixed $\mathbf{y}$ with coprime coordinates this is a $s-1$ dimensional lattice $\Lambda(\mathbf{y})$. Let $N(X)$ denote ...
0
votes
0
answers
144
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Siegel's Mean Value Theorem by Macbeath and Rogers
It is claimed in an answer in mathoverflow to a question about Siegel's Mean value theorem (link- Siegel's Mean Value Theorem by Rogers and Macbeath) that there is mistake for the case $n=2$. I ...
1
vote
0
answers
129
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$L^\infty$ norm lower bound for Integer points in null spaces of recursively defined integer vectors?
Letting $\otimes$ be matrix kronecker/tensor product with $n\in\Bbb N$ as a parameter define non-negative integer vectors recursively $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$
...
5
votes
0
answers
602
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Necessary and Sufficient condition for Sharpness of Bombieri and Vaaler's result on Siegel's lemma?
This Wikipedia page currently quotes Bombieri and Vaaler's result on Siegel's lemma:
Suppose we are given a system of $m$ linear equations in $n$ unknowns such that $n>m$, say
$a_{11}x_1+\dots+...
1
vote
0
answers
45
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Integral matrices with a lot of small integral vectors in the kernel
Suppose $A$ is an $m\times n$ matrix with integer coefficients. These coefficients are possibly very large, however we assume there are at least $K_C$ vectors $x\in\mathbb Z^n$ with $\max_i |x_i|\leq ...
11
votes
4
answers
446
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Sequential addition of points on a circle, optimizing asymptotic packing radius
Suppose I have to put $N$ points $x_1, x_2, \ldots, x_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced ...
1
vote
0
answers
124
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Computing the successive minima of the following lattice in $\mathbb{R}^4$
Let us define the lattice $\Lambda$ in $\mathbb{R}^4$ defined by the matrix
$$
\Lambda = \begin{bmatrix}
A & 0 & 0 & 0 \\
0 & A & 0 & 0 \\
\gamma_1 & \gamma_2 &...
6
votes
2
answers
559
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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 ...
2
votes
0
answers
58
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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
720
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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 ...
4
votes
1
answer
414
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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 \...
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
266
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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
86
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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
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:
$n = a^2 + b^2 + c^2$ ...
6
votes
1
answer
195
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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
3
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
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 ...