# 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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### Infinite series over lattice of Gaussian integers

I'm trying to show that the following sum converges to $0$ over the lattice $L = \mathbb{Z}[i]$ of Gaussian integers: $$140\times\sum_{\substack{l \in L \\ l\neq 0}} l^{-6} = 0.$$ I don't really ...
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### Let $x,y>1$ be coprime integers and $g>0$ a real number such that $g^x,g^y$ are both integers. Is it true that $g\in\mathbb N$?

Let: $x, y\$ be coprime integers greater than $1$ $g \in \mathbb{R}^+$ $g_,^x \ g^y \in \mathbb{N}$ Proposition: $g \in \mathbb{N}$ I have not managed to prove it. Via the fundamental theorem of ...
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### Test if a point on a hexagonal lattice falls on a specified superlattice?

Based on previous answers (1, 2, 3) integers $i, j$ produce a hexagonal lattice using $$x = i + j/2$$ $$y = j \sqrt{3} / 2.$$ From a point $k, l$ I can make a superlattice from integers $I, J$ ...
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### Non-parity based approach to the problem of showing that the area of a convex lattice pentagon is $\ge 5/2.$

I solved the problem given in the title with my approach shown below, but I was wondering whether there's any cleaner solution that does not rely on dividing the vertices of the polygon into the ...
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### $(x,y)$ pairs in lattice $Z^2$ that are co-prime with euclidean-norm at most $k$

Let $B(k) = \{(x,y)\in Z^2 ~|~ x^2+y^2\leq k^2\}$, where $Z$ is the set of integers. It is quite straight forward to show that $|B(k)|$ is $\Theta(k^2)$. My question is whether the number of co-...
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### Indecomposable elements in a lattice

Let $L$ be an discrete lattice in $\mathbb R^n$. We say that a nonzero $a\in L$ is indecomposable if and only if $a$ cannot be written as $a=b+c$ with $b,c$ nonzero and $b^T c>0$. I was initially ...
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### Criteria for checking if points are the vertices of a hypercube

I asked a question over at Code Golf Stack Exchange which essentially asked folks to write a program to determine if a collection of $2^n$ points in $\mathbb{Z}^m$ is the vertex set of some $n$-...
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### Complete Boolean Lattice

What is the formal definition of Complete Boolean lattice $Q_n$? What would be an example for $n=4$, i.e. $Q_4$?
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### Points in a triangular lattice at the same distance from the origin and “breaking of symmetry”

Introduction I was trying to simulate what would happen to a certain physical system taking place in a triangular lattice (the physical details are not relevant to the discussion), when I came across ...
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### Parametrization of all unimodular matrices of rank $n$?

For a given rank $n$, are there parametrization families that cover all possible unimodular matrices of rank $n$? For example for $n=3$ Wolfram gives an example of one parametrization family.
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### Solving the HNP using lattice reduction?

I am trying to understand how lattice reduction can be used to solve the hidden number problem. The question came up after looking into this paper: https://eprint.iacr.org/2019/023.pdf The papers ...
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### Testing for integer points in a parallelepiped

Given an invertible affine transformation $f:\Bbb R^n\to\Bbb R^n$ represented as $f(x)=Ax+b$, I'd like to know whether $f((0,1)^n)\cap\Bbb Z^n$ is nonempty. Is there an efficient way to compute this? ...
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### Determinant of lattice is $q^n$ with high probability?

Lemma: Let $\mathbf{A}\in\mathbb{Z}_q^{n\times m}$ be a uniformly random matrix and $\Lambda^\perp(\mathbf{A})=\lbrace x\in\mathbb{Z}^{m} : \mathbf{A}^Tx\equiv\mathbf{0}\ (\text{mod }q)\rbrace$ be a ...
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### Orthogonal complement to a lattice

Suppose we have an even lattice of rank 2, $\Lambda$, with the following intersection form, \begin{eqnarray} \left( \begin{array}{cc} 2 & 3 \\ 3 & 0 \end{array} \right) \end{eqnarray} As far ...
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### helmholtz equation equipped with dirichlet boundary conditions for a square and the asymptotic behaviour of these corresponding eigenvalues.

Marcus here. The expression $-\nabla^2 u = \lambda u$, where $\nabla^2=\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$, is know as helmholtz equation in two-dimensions. If this ...
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### Is the sum of the reciprocals of the indices of sublattices in $\mathbb{Z}^{n}$ covering $\mathbb{Z}^{n}$ greater than or equal to 1?

For a finite number of sublattices $\Lambda_{i}\subseteq L:=\mathbb{Z}^{n}$, assume that $\cup_{i}\Lambda_{i}=L$. My guess is that $\sum_{i} \frac{1}{\left[L:\Lambda_{i}\right]}\ge 1$. Is my guess ...
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### Exponentially decayed weight lattice path

Consider a 4 dimensional lattice with spacing $\Delta$. Want to get expression of $$\sum_\text{all paths} e^{-k(\Delta) L}$$ in the limit of $\Delta \rightarrow 0$ where $L$ is the number of "links"...
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### Moving an object in a lattice

There's this problem I've been thinking about. Suppose we have a geometric shape (a ball, or a rectangle) of specific size, and we would like to know if it's possible to move it (by translating or ...
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### Number of possible rotational domains of one 2D lattice on top of another?

There are four (or five) two dimensional Bravais lattices which I refer to as oblique, rectangular, hexagonal and square. I'll discuss the 17 symmetry wallpaper groups below. Ignoring translation, ...
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### General solution to this functional equation.

Let $\Lambda_8$ be the $E_8$ lattice. The functional equation is: $$f(a) = \sum\limits_{a=b+c} f(b)f(c)$$ Where $f(a)$ is defined only for points on the lattice $a\in \Lambda_8$ What would be a ...
Let $M \subset L$ be two lattice of $\mathbb{R}^2$ and $|L:M|=2$. Let $v_1,v_2$ be basis of $M$ and linearly independent in $L$. Assume further that $||v_1||\leq ||v_2||$. If we fix $v_1$, can we ...