# Questions tagged [geometry-of-numbers]

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• 143
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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.
• 12.1k
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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, ...
• 22.6k
180 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 ...
• 445
1 vote
92 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
58 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 ...
• 445
1 vote
77 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
186 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 ...
• 445
205 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 ...
• 445
262 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 $i$-...
• 445
70 views

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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 ...
• 1,565
156 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
298 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
185 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 $x-[x]$ ...
• 11.9k
175 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 $\mathbb{Z}^n$, ...
228 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 $n$...
• 9,451
1 vote
212 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
178 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 ...
278 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 ...
• 19.3k
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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 ...
396 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 ...
527 views

• 3,605
1 vote
114 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 ...
• 1,573
267 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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322 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{...
• 3,605
118 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 ...
• 150k
1 vote
91 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 ...
• 215
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: $n = a^2 + b^2 + c^2$ ...
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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 \$\...