Questions tagged [geometry-of-numbers]
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92 questions
3 votes
2 answers
311 views
Shortest vector in orthant of lattice
There are many great algorithms for enumeration of vectors in a lattice such as Fincke-Pohst-Kannan, extreme pruning etc, not to mention great implementations such as fplll. Let $L$ be high density (...
- 273
2 votes
0 answers
57 views
Comparing $i$ th minima of a lattice and discrete subgroup
This is a continuation of the previous question Comparison between first minimum of a lattice and a discrete subgroup in function field. Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}...
- 151
5 votes
0 answers
254 views
Integral polytopes of given genus
Let $n \geq 1$ be an integer. An integral polytope $\Delta \subseteq \mathbb{R}^n$ is the convex hull in $\mathbb{R}^n$ of a finite set of points of $\mathbb{Z}^n$. Two such polytopes $\Delta$ and $\...
- 1,326
0 votes
1 answer
207 views
Principal prime ideals in imaginary quadratic number field
Let $K$ be an imaginary quadratic number field. If the class number of $K$ is greater than 1 there exist non-principal ideals. Some of the ideals that are principal have prime norm, i.e. they are ...
- 273
1 vote
1 answer
228 views
Comparison between first minimum of a lattice and a discrete subgroup in function field
This is the continuation of the previous question Lattices in the extension field of local fields in positive characteristic. Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}_q$, where ...
- 151
3 votes
1 answer
337 views
Lattices in the extension field of local fields in positive characteristic
Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}_q$, where $q$ is a prime power. The norm in this field is defined by $ \left| \frac{f}{g} \right| = q^{\deg(f) - \deg(g)}, $ where $f, ...
- 151
8 votes
0 answers
291 views
Discriminants and lattices in Algebraic geometry vs Geometry of numbers
(Post-writing, this question ended up being way more rambly than I intended. Sorry for that. There's a lot of closely related ideas I'm trying to unravel and it's hard to extract an individual ...
- 513
8 votes
1 answer
391 views
Counting matrices of bounded norm in $\mathrm{SL}_n(\mathbb{Z})$
I'm looking for the asymptotic order of growth of the number of points in algebraic groups, such as $\mathrm{SL}_n(\mathbb{Z})$, of height/norm at most $X$, i.e. all entries are at most $X$ in ...
- 1,292
12 votes
1 answer
303 views
Number of planes generated by integer vectors
For fixed dimension $d$ and large $R$ consider all non-zero integer vectors in the ball $B(0,R)\subset \mathbb{R} ^d$ of radius $R$ centered at the origin. The number of such vectors grows as $c_d\...
- 115k
3 votes
1 answer
601 views
Counting lattice points inside a parallelepiped
The problem I am about to state is in three dimensions and does not follow from Davenport's theorem. Its two-dimensional version is an immediate consequence of Pick's theorem. Consider the lattice $\...
- 464
42 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.
- 13.5k
50 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, ...
- 25.1k
2 votes
0 answers
255 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 ...
- 495
1 vote
1 answer
133 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 ...
- 111
1 vote
0 answers
101 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 ...
- 495
1 vote
0 answers
111 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 ...
- 237
4 votes
2 answers
301 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 ...
- 495
2 votes
1 answer
318 views
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 ...
- 495
6 votes
2 answers
625 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 $i$-...
- 495
0 votes
0 answers
130 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 $\...
- 495
8 votes
2 answers
820 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,\...
- 511
3 votes
0 answers
91 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 ...
- 1,575
4 votes
0 answers
188 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 ...
- 628
2 votes
2 answers
425 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
218 views
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 $x-[x]$ ...
- 12.3k
6 votes
1 answer
302 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 $\mathbb{Z}^n$, ...
9 votes
1 answer
285 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 $n$...
- 9,741
1 vote
1 answer
286 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 ...
- 503
5 votes
1 answer
229 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 ...
user14875
4 votes
1 answer
345 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 ...
- 21.8k
0 votes
1 answer
296 views
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 ...
user147650
6 votes
1 answer
498 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 ...
7 votes
1 answer
680 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\...
- 21.8k
4 votes
2 answers
362 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 $$ w^2 - ax^2 -by^2 + abz^2 = 1 $$ $$ \...
- 2,070
1 vote
1 answer
227 views
Lowering $i$th shortest vector of a lattice
LLL guarantees that we can find a basis $v_1,\dots,v_n$ of a lattice in $\mathbb R^n$ with $$\|v_i\|\leq \gamma_{i,n} \det(\Lambda)^{1/(n-i+1)}$$ where $\gamma_i$ is a function only of $i$ and $n$. ...
- 1
2 votes
1 answer
278 views
Which lattices are rotatable into their scaled copy?
Let $L=\{\sum_i n_iv_i\mid n_i\in\mathbb Z\}$ be some lattice generated by $d$ independent vectors $(v_i)_1^d$ from $\mathbb R^d$. Call $L$ rotatable if for some $M$, a scalar multiple of some ...
- 19.7k
9 votes
0 answers
508 views
Number fields ordered by discriminant
Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are ...
- 13.9k
15 votes
1 answer
723 views
Counting primitive lattice points
In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result): Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then $$\# \...
- 22k
-1 votes
2 answers
266 views
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 $$N=...
- 1
2 votes
1 answer
379 views
Intuition behind the proof of key step in Minkowski's second inequality on successive minima
I recently knew of this note in which Prof. M. Henk presents a proof of Minkowski's second inequality on successive minima which is (purportedly) based on ideas in Minkowski's original proof. Let me ...
- 9,097
5 votes
0 answers
162 views
Averaging number of lattice points in a box over a family of lattices
Consider the diophantine equation $$ x_1y_1^3 + \dots + x_s y_s^3 = 0. $$ For fixed $\mathbf{y}$ with coprime coordinates this is a $s-1$ dimensional lattice $\Lambda(\mathbf{y})$. Let $N(X)$ denote ...
- 163
0 votes
0 answers
174 views
Siegel's Mean Value Theorem by Macbeath and Rogers
It is claimed in an answer in mathoverflow to a question about Siegel's Mean value theorem (link- Siegel's Mean Value Theorem by Rogers and Macbeath) that there is mistake for the case $n=2$. I ...
- 111
1 vote
0 answers
166 views
$L^\infty$ norm lower bound for Integer points in null spaces of recursively defined integer vectors?
Letting $\otimes$ be matrix kronecker/tensor product with $n\in\Bbb N$ as a parameter define non-negative integer vectors recursively $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$ ...
- 1
5 votes
0 answers
657 views
Necessary and Sufficient condition for Sharpness of Bombieri and Vaaler's result on Siegel's lemma?
This Wikipedia page currently quotes Bombieri and Vaaler's result on Siegel's lemma: Suppose we are given a system of $m$ linear equations in $n$ unknowns such that $n>m$, say $a_{11}x_1+\dots+...
- 1
1 vote
0 answers
78 views
Integral matrices with a lot of small integral vectors in the kernel
Suppose $A$ is an $m\times n$ matrix with integer coefficients. These coefficients are possibly very large, however we assume there are at least $K_C$ vectors $x\in\mathbb Z^n$ with $\max_i |x_i|\leq ...
- 131
11 votes
4 answers
571 views
Sequential addition of points on a circle, optimizing asymptotic packing radius
Suppose I have to put $N$ points $x_1, x_2, \ldots, x_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced ...
- 5,999
1 vote
0 answers
147 views
Computing the successive minima of the following lattice in $\mathbb{R}^4$
Let us define the lattice $\Lambda$ in $\mathbb{R}^4$ defined by the matrix $$ \Lambda = \begin{bmatrix} A & 0 & 0 & 0 \\ 0 & A & 0 & 0 \\ \gamma_1 & \gamma_2 &...
- 3,773
6 votes
2 answers
902 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 ...
- 3,773
2 votes
0 answers
75 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 ...
- 21
8 votes
1 answer
899 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 ...
- 201