Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [geometry-of-numbers]

The geometry of numbers studies convex bodies and lattice points.

43 questions
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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 in Diophantine Geometry, which is a deep theorem…
James Moriarty
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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 $d$-dimensional ball. The paper cites a German book Einführung…
Guy
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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 $\vec{v_1}$ and $\vec{v_2}$ are linearly independent.…
tc1729
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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{2}, \sqrt{3}) = \mathbb{Q}[x,y]/(x^2 - 2, y^2 -…
cactus314
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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 $K$. These are examples of lattices in $K$, which are…
Pedro
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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: Since circle is about origin, hence, integral…
Aang
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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-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}}…
user02138
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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 triples and you get 3 points. and i am stuck with 3…
endgame yourgame
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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 lattice points are not necessarily integer, i.e.,…
M.X
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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 $\displaystyle[z^{n^2}]\frac{1}{1-z}\Theta(z)^d$ where…
Brian Burns
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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 considering a rectangle. I don't know how to do…
mtheorylord
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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 $\gamma_n = \sqrt{\frac{4}{3}}$, and this value is…
Chris
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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?
limepickle
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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. Suppose the triangle contains exactly 74 lattice…
Maxwell
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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 $(v_1,\cdots,v_k) \in \mathbb R^{dk}$ such that…
taylor
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