# 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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### Can every lattice obtained by orthogonal projection of the cubic lattice?

Assume that $\Lambda\subset\mathbb{Z}^n$ is a full-dimensional lattice with the generator matrix $G\in\mathbb{Z}^{n\times n}$. Can I always embed $\mathbb{R}^n$ into a bigger dimensional vector space, ...
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### Number of lattice points $7$ units away

A lattice point is a point in space which has all coordinates as integers. How many $3D$ lattice points are exactly $7$ units from the origin? I'm thinking the answer to this is $3$, since we have ...
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### Poisson summation formula for lattices not necessarily of full rank

Let $\Lambda$ be an integral lattice (i.e. $\Lambda \subset \mathbb{Z}^n$) in $\mathbb{R}^n$. If $\Lambda$ is of full rank, i.e. the rank of $\Lambda$ is $n$, and $\phi:\mathbb{R}^n \to \mathbb{R}$ is ...
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### Path Lengths and Random Walk - Generalization of the Uncut Spaghetti Game

The following is a kind of generalization of the interesting Uncut Spaghetti game (https://mathpickle.com/project/uncut-spaghetti-number-patterns/): Consider a 2D integer lattice where each point has ...
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### What is the degeneracy of the "Coxeter plane" for the E8 lattice?

The 240 minimal vectors (roots) of the E8 lattice, projected onto "the" Coxeter plane, are shown here: https://en.wikipedia.org/wiki/E8_(mathematics)#/media/File:E8Petrie.svg and discussed ...
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### Lattice embeddings in $\mathbb{Z}^n$

I have a certain class of lattices, and given $\Lambda$ in this class I would like to find obstructions to the existence of a finite-index lattice embedding $\Lambda \rightarrow \mathbb{Z}^n$ for some ...
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### The number of closed paths in the square lattice $\mathbb{Z}^2$ with length $n$ and starting and ending points at $(0,0)$.

I'm thinking about this problem right now. Problem:Consider a lattice point consisting of $\mathbb{Z}^2$ points. If $n$ is even, i.e., $n=2p$, then Show that the number of closed paths in the square ...
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### Semi-lattices whose Hasse Diagrams are trees after transitive reduction?

Is there a name or anything else known for a semi-lattice whose Hasse Diagram becomes a tree after applying transitive reduction? Trying to find more about it since it comes up in an optimization ...
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### Is there a rational point in a given open set such that the distance from given rational points to it are all irrational numbers on $\mathbb{R}^2$?

On $\mathbb{R}^2$ there is a nonempty open set $A$ and $n$ rational points $a_1,\ldots,a_n$. Is there a rational point $a$ in $A$ such that $\forall i\in\{1,\ldots,n\}, |a−a_i|$ is an irrational ...
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### Is there a rational point in a given open set such that the distancse from given rational points to it are all rational numbers on $R^2$?

On $R^2$ there is a nonempty open set $A$ and n rational points $a_1,...,a_n$. Is there a rational point $a$ in $A$ such that $\forall n\in\{1,..,n\},~|a-a_i|$ is a rational number. My idea is to find ...
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### Why is the covolume of a lattice is unique?

Given a basis $\mathfrak{B}=\{b_1, b_2, \cdots, b_n\}$ of $\mathbb{R}^n,$ the lattice generated by $\mathfrak{B}$ is the set of all linear combinations with integer coefficients: m_1b_1+m_2b_2+\...
Two lattices $\Lambda$ and $\Omega$ (the $\mathbb{Z}$-span of a linearly independent set $B\subset \mathbb{R}^n$) are said to be similar if there exist a real orthogonal $n \times n$ matrix $A$ and a ...