# Tagged Questions

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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### Real dimension of $\mathbb{Z}^d \otimes \mathbb{R}$

I have following, probably really trivial question. Lets take $\mathbb{Z}^d \otimes \mathbb{R}$. Consider this as vector space over $R$ by defining $\mu (v\otimes \lambda) = v \otimes (\lambda \mu)$ ...
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### Inverse Gauss circle problem

The Gauss circle problem focuses on the amount of lattice points of a square lattice that can fit inside a circle with radius r. http://i.stack.imgur.com/y4Yf2.png But say I want to fit N lattice ...
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### What is an ideal lattice?

Can anyone describe in layman's terms what an ideal lattice is? I've seen them mentioned in many places, but haven't found a good definition of what exactly they are, nor any good terms to know where ...
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### Index of a sublattice

Assume that $\Lambda \subset \mathbb{Z}^2$ is an integer lattice of rank $2$ and define for each integer $k$, $$\Lambda(k):=\Big\{(x_1,x_2) \in \Lambda: k\mid (x_1,x_2)\Big\}.$$ What is the index of ...
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### An inequality related to lattice points 'around' a circle

Take a circle of radius $r$ with centre at the origin such that $r^2=N_1^2+N_2^2$ for $N_1,N_2\in\mathbb{N}$. Consider a lattice coordinate $(a,b)$ such that $a\in(-r,-2)$ and define $b$ to be the ...
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### Interpreting infinite integer lattice as a manifold of negative dimension

Various fractal dimensions coincide on self-similar fractals as the logarithm of self-copies the fractal includes divided by the logarithm of the factor by which the copies are smaller than the ...
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### Finding total number of lattice points on a circle and one of the coordinate values

I was wondering if it was possible to quickly find the total amount of lattice points on a circle, given its equation (origin and radius), and I was also a bit confused on how to find a particular ...
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### Maximize the product of pairwise distances between particles on a lattice

I am trying to solve the following problem : Consider an $N\times N$ square lattice ($N$ even integer, we can assume that N is large) on the complex plane, with lattice sites at position $j+ik$, $j$ ...
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### Number of Lattice points in the intersection between an N-Sphere and an N-Cube

I have a hyper-Cube lattice with coordinates between 0 and 10, therefore 11^N lattice points. I take one of this point, C, and then another point P at distance D from C. I need to know a good ...
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### Compute a reduced basis for the lattice with basis matrix

For an exercise I need to compute a reduced basis for the lattice with basis matrix: \begin{pmatrix} 1 & 0 \\ 1414 & 1000 \\ \end{pmatrix} I found the algorithm for ...
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### Ideals of $\mathbb{Z}[i]$ geometrically

It is pretty easy to visualize the ideals of $\mathbb{Z}$ in the "integer line". Let's go up to $\mathbb{Z}[i]$ and consider the ideal $3\cdot\mathbb{Z}[i]$. We can visualize it as a "sub-lattice" ...
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### Finding the number of lattice paths

Find the number of lattice path of length $2n$ that starts on $(0, 0)$ such that for all the points $(x, y)$ in the path, $x < y$. So pretty much all the points besides the origin are strictly ...
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### n-dimensional lattice as a collection of lower dimensional spaces.

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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### Covering a family of sets of $\mathbb{Z}^d$ with boxes of a given diameter

Let $diam(A)$ be the graph distance between the two farthest vertices contained in a finite set $A \subset \mathbb{Z}^d$. Is it true that there exists a real $\eta(d)>0$ such that for any finite ...
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### Why are there only so many Bravais Lattices?

I am in doubt as to why there are exactly five 2d Bravais lattices? For example, I could take the square lattice and place a lattice point at the midpoint on every side of each square. Shouldn't ...