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Questions tagged [geometry-of-numbers]

The geometry of numbers studies convex bodies and lattice points.

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How to compute the volume of a high-dimensional cube with complex and dependent coordinates?

I am reading a paper on ideal lattices: On page $15$ ( in the proof of Lemma $6.1$ ), it says: The cube $\mathcal{C}_2 = \left\{\left(z,\bar{z}\right) \in \mathbb{C}^{2} : \left\vert z \right\vert \...
Han's user avatar
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Seeking an elementry theorem about lattices.

I am doing some work with objects. Each object has a corresponding embedded free $\mathbb Z$-module, with important properties of the object being related to whether the embedding is a lattice. From ...
Sriotchilism O'Zaic's user avatar
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Natural extension of Gauss measure (continued fractions)

There's a Gauss measure on $[0,1]$ given by its density $$f(x)=\frac 1{\ln 2} \frac{1}{1+x} $$ It's invariant under Gauss map $x\mapsto\big\{\frac 1 x\big\}$ which acts as a left shift if we describe ...
Big Coconut's user avatar
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Number of lattice points of a convex body transformed by a matrix

Let $\boldsymbol{A}=[\boldsymbol{a_1},\boldsymbol{a_2},\ldots,\boldsymbol{a_T}]^\text{tr}\in\mathbb{R}^{T\times D}$, and $S$ be the integer lattice points within and on the surface of a convex body. ...
MohammadJavad Vaez's user avatar
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Probability that an integer polynomial factors in a particular family

How can I estimate $$ \Pr\left(f(x)=\sum_{i=0}^na_ix^i\text{ factors over }\mathbb{Q} \ \Bigg| \ \sum_ia_i=N, \ 0\leq a_i\in\mathbb{Z}\right)? $$ How about for fixed $n$ and $N\to\infty$? The ...
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Explicit bounds for counting integer points inside $2$-norm balls

It is well known that a ball of $2$ norm radius $R$ in $d$ dimensional space constains \begin{align} N(R) = V_d R^d + \mathcal{O}(R^{d-2}) \end{align} points with integer coordinates (i.e. ...
ors's user avatar
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Upper Bound on the shortest non-zero non-negative vector in an integer lattice

Suppose we have an integer lattice of full rank $L\subseteq\Bbb{Z}^n$, say of determinant $D$. Do we have a nice bound on the smallest non-zero vector $x\in L$ s.t. $x_i\geq0$ for all $i$? Perhaps ...
yotam maoz's user avatar
3 votes
1 answer
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Bound on the number of bounded primitive $k$-tuple for all unimodular lattices

Let $X_d$ be the set of unimodular lattices in $\mathbb R^d$ (lattices with covolume one). For a lattice $\Lambda$ in $X_d$ and $1\le k \le d$, let $P^k(\Lambda)$ denote the set of all $k$-tuples $(...
taylor's user avatar
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Distribution of lattice points visible from the origin

I am reading Apostol's book on Introduction to Analytic Number Theory. In section 3.8, he discusses the distribution of lattice points visible from the origin. While finding the density of such points ...
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Quadratic forms: Existence of $(x,y)\in \mathbb{Z}^2 \setminus \{0\}$ such that $P(x,y) < 2\sqrt{\lvert \det(P) \rvert}$

I am stuck on the following exercise: Show that for any non-degenerate quadratic form $P$ over $\mathbb{R}$, that is either indefinite or positive definite, exists an integer point $(x,y) \in \mathbb{...
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3 votes
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Calculating the Minkowski bound of $\mathbb{Z}[\sqrt{8}]$

The formula for the Minkowski bound is $$M=(\frac{4}{\pi})^{s}\frac{n!}{n^n}\left| \Delta \right|^{\frac{1}{2}}$$ where $s$ is the number of pairs of complex embeddings. In our case, $s=0$, $n=2$, $\...
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Counting $S$-Integer Lattice Points

The fundamental principle underlying the geometry of numbers is that the number of lattice points in a round, compact subset $R \subset \mathbb{R}^n$ is roughly given by the volume of $R$. A very ...
Ashvin Swaminathan's user avatar
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Geometry of $\mathcal{O}_K/\mathcal{O}_K^\times$

Let $K$ be a number field, $\rho_1,\dots,\rho_{r}$ its real embeddings, and $\sigma_1,\bar{\sigma_1},\dots,\sigma_{s},\bar{\sigma_{s}}$ its complex embeddings. There's a map \begin{align*} K &\...
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Existence of $e$ such that there exist integers $x,y$ not both $0$ and that $\left|ax^2 + bxy + cy^2\right|\le e\sqrt{b^2 - 4ac}$

Let $a,b,c$ be real numbers with $a > 0$. Suppose that $d = b^2 - 4ac > 0$. Is there a constant $e$ such that there exist integers $x,y$ not both $0$ and that $\left|ax^2 + bxy + cy^2\right|\le ...
user0's user avatar
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1 answer
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How to find vertices of a $5$-dimensional simplex, where the vertices are formed by zeroes and ones?

I'm trying to find the vertices of a simplex in the $5$-dimensional space, where the vertices are formed by only zeros and ones, similarly to these coordinates that represent a simplex in the $7$-...
Nillmer's user avatar
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2 answers
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How to construct six points $ABCDEF$ on a plane so that the distance between any two of them is an integer, and no three are collinear?

How to construct six points $ABCDEF$ on a plane so that the distance between any two of them is an integer, and no three are collinear? I tried with some right angled triangles with pythagorean ...
endgame yourgame's user avatar
1 vote
0 answers
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Minkowski theory: an isomorphism of $\Bbb R$-vector spaces induces a scalar product on $\Bbb R^{r + 2s}$

I'm following Neukirch's algebraic number theory. The situation is as follows: Let $K$ be a number field of degree $n$. Then $n = r + 2s$, where $r$ is the number of real embeddings $\rho : K \to \...
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Understanding a crude estimate for the number of lattice points inside a ball

I've been reading the appendix (A lattice sum) of this write up by Keith Conrad and pretty much understand most of the argument but I'm stuck on the following: Let $S(x)$ denote the number of non-...
1729's user avatar
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21 votes
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In Search Of Elementary Proof Of Kobayashi's Theorem

There is a theorem in Number Theory due to Hiroshi Kobayashi (possibly less famous). The statement of this theorem is quite simple-looking. The original proof of Kobayashi relies on Siegel's Theorem ...
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4 votes
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Find $n$ points on a circle with integer distances.

Let $n$ be a positive integer, prove that it is possible to put $n$ points on a circle so that the distances among them are all integers. For $n \leq 3$ this is trivial. I have shown it for $n=4$ by ...
mtheorylord's user avatar
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9 votes
2 answers
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Solve $a^2 - 2b^2 - 3 c^2 + 6 d^2 =1 $ over integers $a,b,c,d \in \mathbb{Z}$

Are we able to completely solve this variant of Pell equation? $$ x_1^2 - 2x_2^2 - 3x_3^2 + 6x_4^2 = 1 $$ This has an interpretation as is related to the fundamental unit equation of $\mathbb{Q}(\sqrt{...
cactus314's user avatar
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4 votes
1 answer
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Proof for known values of the Hermite constant

I understand that the values of the Hermite constant for $1 \leq n \leq 8$ and $n=24$ have been determined exactly. For example, Lagrange proved for $n=2$ the value of the Hermite constant is $\...
Chris's user avatar
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Can interesting bounds to Gauss circle problem be seen/come from counting points close to a line?

Let $N(r)$ be the number of lattice points distance at most $r$ from the origin in $\mathbb{R}$. The Gauss circle problem is a famous problem which is looking to understand the error term $E(r):= N(r)-...
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1 vote
1 answer
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Lagrange's theorem via Minkowski's first theorem

When a given author has developed the basics of the geometry of numbers, he/she usually proceeds to exemplify the theory by proving Lagrange's theorem on the representability of the natural numbers as ...
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Visualize ideals in number fields

Let $K$ be a field extension of degree $n$ over $\mathbb{Q}$. We know that the ring of integers $\mathcal{O}_{K}$ is a free $\mathbb{Z}$-module of rank $n$, and so is any fractional ideal $I$ in $K$. ...
Pedro's user avatar
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2 votes
1 answer
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Reference for Noam Elkies comments on integral lattices and fundamental parallelotope

ORIGINAL: I would like to find some more detailed references to explain 2012 comments of Prof. Noam Elkies at https://mathoverflow.net/questions/103152/determinant-of-integer-lattice-basis-of-l-x-1-...
Will Jagy's user avatar
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3 votes
1 answer
127 views

discrete subrings of $\mathbb{R}^r\times\mathbb{C}^s$

Given a number field $K$ of degree $r+2s$, its ring of integers $\mathcal{O}_K$ is a discrete subring of $\mathbb{R}^r\times\mathbb{C}^s$ under the diagonal embedding $$ \alpha\mapsto(\sigma_i(\alpha))...
yoyo's user avatar
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3 votes
1 answer
317 views

Generalization of Minkowski theorem [closed]

Can you help proving this generalization of the Minkowski theorem ? Let $\Omega \subset \mathbb{R}^N$ be a nonempty centrally symmetric convex subset. a) Then $\#(\Omega \cap \mathbb{Z}^N) \ge 2 (\...
Andy's user avatar
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1 vote
1 answer
213 views

$\mathbb{R}^d / \Gamma$ where $\Gamma$ is an irrational lattice

A lattice $\Gamma \subset \mathbb{R}^d$ is said to be rational if for any two vectors $\gamma_1, \gamma_2 \in \Gamma$, their inner product satisfies the relation $$\displaystyle \langle\gamma_1, \...
user363087's user avatar
  • 1,145
2 votes
1 answer
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How to post an idea to see if it has any merit?

I am a "hobbyist mathematician", and I have an idea that relates Prime numbers, rational numbers, polytopes in n-dimensional lattices, and lattice point counting in those polytopes. For example, it ...
D.Fay's user avatar
  • 31
3 votes
2 answers
696 views

A proof of Minkowski's ramification theorem without the use of geometry of numbers

Consider the following theorem due to Minkowski (originally conjectured by Kronecker iirc): For any proper finite extension $K$ of $\mathbb Q$, some prime in $\mathbb Q$ ramifies in $K$. ...
Wojowu's user avatar
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3 votes
1 answer
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Minkowski's convex body theorem and binary quadratic forms

Saw this question in NZM and have a lot of difficulties trying to start. Consider the binary quadratic form $f(x,y) = ax^2 + bxy + cy^2$ with $a > 0$ and $d = b^2 − 4ac < 0$. Show that there ...
Noob4398's user avatar
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3 votes
2 answers
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Prove that there does not exist a $n$-regular polygon $(n\ge 4)$, such that its sides and diagonals are all integers.

Prove that there does not exist a $n$-regular polygon $(n≥4)$, such that its sides and diagonals are all integers. Maybe a famous problem, but I don't know how to solve that.
piteer's user avatar
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4 votes
2 answers
164 views

If $(a,b,M)$ is a Pythagorean triple, can $(b,b+a,N)$ be another triple?

Does anyone know of a pair of Pythagorean triples of the form $$(a, b, M) \quad\text{and}\quad(b, b+a, N)$$ Is such a pair possible?
limepickle's user avatar
8 votes
1 answer
2k views

The number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...
Guy's user avatar
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3 votes
2 answers
337 views

Sum of areas of triangles which have corners which are lattice points with 74 lattice points inside.

Here is a problem I was given: A lattice point in the plane is a point whose coordinates are both integers. Consider a triangle whose vertices are lattice points (0,0),(a,0) and (0,b), where b≤a. ...
Maxwell's user avatar
  • 405
5 votes
1 answer
2k views

number of lattice points in an n-ball

I have faced a problem in my work and I will appreciate any hint/reference as I am not much into the lattice problems. Assume an n-dimensional lattice $\Lambda_n$ with generator matrix $G$. Note that ...
M.X's user avatar
  • 713
4 votes
1 answer
296 views

Generating function for lattice points in a sphere

This is a note in Sedgewick's Analytic Combinatorics: The number of lattice points with integer coordinates that belong to the closed ball of radius n in d-dimensional Euclidean space is $\...
Brian Burns's user avatar
7 votes
2 answers
2k views

Fermat's 2 Square-Like Results from Minkowski Lattice Proofs

Minkowski's Convex Body Theorem for lattices in the plane: Suppose $\mathfrak{L}$ is a lattice in $\mathbf{R}^2$ defined as $\mathfrak{L}=\{m\vec{v_1}+n\vec{v_2}:m,n\in\mathbf{Z}\}$, where $\vec{v_1}$ ...
tc1729's user avatar
  • 3,057
1 vote
2 answers
3k views

Counting lattice points interior to a polygon

If I define an integer lattice $\Lambda \subseteq \mathbb{Z}^2$ with a basis given by $$\omega_{1} = a \hat{i} + b\hat{j}, \;\;\; \omega_{2} = -b \hat{i} + a\hat{j}$$ How can I count how many lattice ...
Samuel Reid's user avatar
  • 5,092
8 votes
2 answers
256 views

Radius either integer or $\sqrt{2}\cdot$integer

Given a circle about origin with exactly $100$ integral points(points with both coordinates as integers),prove that its radius is either an integer or $\sqrt{2}$ times an integer. What my solution is:...
Aang's user avatar
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2 votes
2 answers
1k views

Does $Ax + By = C$ pass through any lattice point?

Given an equation of a straight line of form $Ax + By = C$. where $A,B,C$ are integers. How could we check if this passes through any lattice point or not? Please suggest me a suitable algorithm.
user avatar
7 votes
2 answers
893 views

Counting Lattice Points with Ehrhart Polynomials

Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the origin and the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-...
user02138's user avatar
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