Questions tagged [geometry-of-numbers]

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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 ...
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  • 1,441
4 votes
0 answers
129 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 ...
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  • 598
2 votes
2 answers
231 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 ...
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1 vote
1 answer
179 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 $x-[x]$ ...
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6 votes
1 answer
111 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 $\mathbb{Z}^n$, ...
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9 votes
1 answer
212 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 $n$...
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1 vote
1 answer
136 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 ...
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  • 425
4 votes
1 answer
104 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 ...
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  • 447
4 votes
1 answer
219 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 ...
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1 answer
235 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 ...
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6 votes
1 answer
305 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 ...
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5 votes
1 answer
348 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\...
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4 votes
2 answers
261 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 $$ w^2 - ax^2 -by^2 + abz^2 = 1 $$ $$ \...
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2 votes
1 answer
114 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 $n$. ...
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2 votes
1 answer
135 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 ...
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  • 17.1k
8 votes
0 answers
268 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 ...
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15 votes
1 answer
412 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 $$\# ...
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-1 votes
2 answers
226 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=...
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  • 12.8k
2 votes
1 answer
271 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 ...
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5 votes
0 answers
122 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 ...
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  • 163
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0 answers
126 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 ...
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1 vote
0 answers
128 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 $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$ ...
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  • 12.8k
5 votes
0 answers
592 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+...
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  • 12.8k
1 vote
0 answers
41 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 ...
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11 votes
4 answers
420 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 ...
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  • 5,758
1 vote
0 answers
113 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 &...
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  • 2,463
4 votes
2 answers
343 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 ...
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  • 2,463
2 votes
0 answers
53 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 ...
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  • 21
8 votes
1 answer
588 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 ...
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  • 201
1 vote
0 answers
112 views

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 ...
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  • 1,369
6 votes
0 answers
230 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 ...
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  • 5,758
2 votes
3 answers
318 views

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{...
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  • 2,463
6 votes
0 answers
108 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 ...
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1 vote
1 answer
73 views

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 ...
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  • 211
4 votes
2 answers
4k 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: $n = a^2 + b^2 + c^2$ ...
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  • 21.8k
6 votes
1 answer
175 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 $\...
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  • 86.7k
2 votes
2 answers
1k views

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 ...
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  • 497
5 votes
1 answer
566 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 ...
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  • 497
3 votes
0 answers
180 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, ...
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  • 497
5 votes
1 answer
302 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, ...
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  • 509
7 votes
0 answers
184 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,...
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  • 1,292
7 votes
2 answers
237 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 ...
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  • 91
2 votes
0 answers
383 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 ...
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  • 779
2 votes
1 answer
243 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 ...
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3 votes
1 answer
781 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_{ ...
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  • 51
5 votes
1 answer
461 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 ...
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  • 93.7k
3 votes
1 answer
404 views

n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
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  • 31
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 ...
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15 votes
1 answer
582 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 $S$...
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  • 8,175
6 votes
3 answers
637 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 $A \in M_n(\mathbb Z)$ ...
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