# Questions tagged [geometry-of-numbers]

The geometry of numbers studies convex bodies and lattice points.

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• 579
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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 ...
• 97
1 vote
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• 55
1 vote
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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 ...
• 2,716
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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 &\...
• 168
1 vote
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• 4,598
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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-...
• 2,157
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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 ...
• 6,114
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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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• 678
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• 395
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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 $\vec{v_1}$ ...
• 3,057
1 vote
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 ...
• 5,092
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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:...
• 14.7k
### 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.
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-...