# Questions tagged [geometry-of-numbers]

The geometry of numbers studies convex bodies and lattice points.

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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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### 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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### 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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### 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:...
### 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-...