# 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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### Lattice width of $conv(0,ne_1,\cdots,n e_n)$

Given a subset $K \subseteq \mathbb{R}^n$ we define the lattice width of $K$ : $$\omega(K) = \min_{d\in \mathbb{Z}^n - \{0\}} \max_{x,y \in K} d^t(x-y)$$ With $K = conv(0,ne_1,\cdots,n e_n)$ how to ...
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### Shortest vector problem as hidden subgroup problem

I posted this question on the cryptography stack exchange with a bounty, but I haven’t gotten much attention. I think part of the reason might be that I’m really interested in the use of group theory ...
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### Predicting the bottom right entry in the output QR decomposition

The following may sound like a very niche question, but a good answer would advance lattice reduction. See details below. As an example, consider the matrix A = \begin{bmatrix} .111 & .025 & ....
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### Which faces does "sphere" lattice polyhedron $\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq4^a(8b+7)$ have?

$\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq 4^a(8b+7)$ is a lattice polyhedron. By Legendre's three-square theorem, such $n$ have representation(s) as the sum of $3$...
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### Extending a closest point to a basis for a lattice

We are given a lattice $\Lambda$ in $\mathbb{R}^2$, and we are given that it contains a basis of 2 vectors, such that the integer combinations span the whole space. Now we choose some $u$ in $\Lambda$,...
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### Number of lattice points of a convex body transformed by a matrix

Let $\boldsymbol{A}=[\boldsymbol{a_1},\boldsymbol{a_2},\ldots,\boldsymbol{a_T}]^\text{tr}\in\mathbb{R}^{T\times D}$, and $S$ be the integer lattice points within and on the surface of a convex body. ...
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### Showing that the lattice D_4^+ is congruent to Z^4

I'm currently reading Conway and Sloane's Sphere Packings, Lattices and Groups. It is a great read for facts, but they only sketch proofs at a high level. Within it, they define the lattice $D_n$ to ...
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### Relation between volume of a lattice and the number of integer points in this lattice

In "Algebraic Number Theory and Fermat's Last Theorem" by Stewart & Tall the volume is, as usually, defined as an integral. Now, exercise 6.1 (page 138) asks to show that the volume of a ...
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### Matrix and Lattice Paths

I have a $k\times k$ matrix  A_{k}= \begin{pmatrix} 1 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & \cdots & 1 &1 & 0\\ &\vdots & &\vdots \\ 1 & 1 &...
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### Number of Lattice Paths with no Loops

Consider an $n$-dimensional hypercubic lattice, where the set of vertices is $\mathbb{Z}^n$. Let $E$ be the union of all edges between adjacent points (i.e. the grid lines). Given a sequence of ...
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### In what sense is the geometry of the dual lattice "reciprocal" to the original lattice?

I've always struggled to get good intuition for how the geometry of the dual lattice relates to the geometry of the original lattice, even in the Euclidean case! Looking for more sources on this, I ...
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### Lattice Desargues Configuration

Here's a Desargues Configuration with all points on a square lattice: Is there a more compact version, either on a square lattice or triangular lattice? Also, if there a less obvious Desargues ...
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### Integer vector subsets invariant under rational matrix

Let $A$ be an $n\times n$ matrix with rational coefficients. Define $H$ to be the maximal subset of $\mathbb{Z}^n$ such that $AH\subset H$. Question: How to describe $H$ in terms of $A$ (its Jordan ...
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### Properly defining a subset of $\mathbb{Z}^k$

I need to provide a good definition of the subset of $\{\{0,1,\ldots,n-1\} \times \{0,1,\ldots,n-1\} \times \cdots \times \{0,1,\ldots,n-1\}\} \subset \mathbb{Z}^k$ given by all the vertices of the ...
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### Making $\{0,\dots,9\}^2$ from smallest subset with coordinate-wise min and max

(Iranian Combinatorics Olympiad-2020) Consider the set of $100$ ordered pairs $A = \{(0, 0), (0, 1), \dots, (9, 8), (9, 9)\}$. Given any subset $S$ of $A$, we have a device to append more elements to ...
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### Enclosing a lamp in space via Minkowski's theorem

I'm currently working on a problem on the chapter of Geometry and Numbers from Andreescu and Dospinescu's Problems from the Book (highly recommended to read). The problem statement is the following: ...
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### Are there any six-dimensional lattices with kissing number equal to 36 or 24?

I am studying two lattices which seem to have kissing number 36 and 24, respectively, and I am curious if they are some known lattices in the literature. I have looked up Conway and Sloane but didn't ...
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### Catalan Numbers: Sequence of -1 and 1 that sums to 0 with conditions

There is a relatively simple bijection between 0-sum sequences of 1 and -1 where the sum of all partial sequences is nonnegative and Dyck paths - this is very easy to count as a Catalan number. ...
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