# Questions tagged [integer-lattices]

A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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### Finding the closest point in a root lattice

Let $L_n$ be a crystallographic root lattice, embedded inside $\mathbb{R}^n$. This means that $L_n$ is the $\mathbb{Z}$-span of the simple roots $\alpha_1, \ldots, \alpha_n \in \mathbb{R}^n$, which ...
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### Square root “base”

In normal base system (say binary), you'll do P = [1,2,4,8, ...] then a value can be represented as coefficients $\{a_i\}$ where $$N=\sum_{i=0}^n a_i\cdot P_i$$ ...
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### Circle-to-circle contacts on a hexagonal grid.

In OEIS sequences A047932 and A263135, coins are placed in a "spiral" on the faces or vertices of a hexagonal grid respectively, and the number of coin-to-coin contacts are counted. The ...
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### Why $\mathbb Z(\sqrt2)$ is not a lattice?

The set $\mathbb Z(\sqrt2) = \{a + b\sqrt2 : a, b \in \mathbb Z\}$ is not a lattice, according to the book of Robeldo = because when you replace $a, b \in \mathbb Z$ by $a, b \in \mathbb R$ we do ...
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### Conditions under which the Lattice Generated by a Subset of Lattice Vectors is Equivalent to the Original Lattice

In general, the sublattice $L_2$ generated by subset of vectors of $L_1$ need not have the same rank as $L_1$. Even if it does, it may be a proper sublattice of $L_1$. However, if the rank and ...
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### Visibility of points on a rectangular lattice under a certain distance from the origin

If I have points a rectangular lattice such that each point takes the form of $(a+bi,c+dj)$ with $i,j\in \Bbb Z$ (for some given $a,b,c,d\in\Bbb Z$), and $b,d>0$. How can I find the number of ...
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### Transforming computable numbers into analytic functions using diophantine equations?

By Matiyasevich's theorem, any enumerable set $\mathbf{S}$ can be expressed as solutions of a diophantine equation in the integers. These can always be expressed as degree 4 polynomials in the ...
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### How many positive integral solutions exist for $2a+3b-c = 0$ where $a$ ranges from $0$ to $5$, $b$ from $0$ to $10$ and $c$ from $0$ to $40$?

I was stuck with this particular problem. I tried finding a solution by attempting to find the coefficient of $x^0$ in $(1+x^2 +\dots+x^{10})(1+x^3 +\dots+x^{30})(1+x^{-1} +\dots+x^{-40})$ but for ...
### The canonical metric on the space of lattices in $\mathbb R^n$
Let $\Lambda_1, \Lambda_2$ be two lattices on $\mathbb R^n$, (is there/)what is a canonical way to define the distance between $\Lambda_1, \Lambda_2$ such that the metric gives the topology on the ...
Given a lattice $\mathcal{L}$ with minimum $\lambda_1(\mathcal{L})$, how can we describe the minimum of a translated lattice $t + \mathcal{L}$ for some $t \in \text{span}(\mathcal{L})$, \$t \notin \...