Questions tagged [geometry-of-numbers]

The geometry of numbers studies convex bodies and lattice 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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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 ...
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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 ...
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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 $7$-...
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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 ...
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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-...
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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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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 ...
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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{...
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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 $\...
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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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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 $K$. ...
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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-...
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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))...
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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 $\#(\Omega \cap \mathbb{Z}^N) \ge 2 (\...
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$\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, \...
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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 ...
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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 $K$. ...
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2 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 ...
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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.
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4 votes
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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?
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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 $d$-...
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286 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. ...
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1 answer
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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 ...
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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 $\...
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5 votes
2 answers
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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 $\vec{v_1}$ ...
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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 ...
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8 votes
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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:...
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2 votes
2 answers
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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.
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7 votes
2 answers
784 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-...
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