# Tagged Questions

Lattices as they are used in number theory. (Not to be confused with lattice theory or lattices as used in physics!)

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### Average range of Motzkin path

Motzkin path are paths from (0,0) to (n,0) in $\mathbb{Z}^2$ such that we are allowed to move SE, E and NE. More on this is here https://en.wikipedia.org/wiki/Motzkin_number I would like to know if ...
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### Intuition behind the definition of the Siegel-Eichler transformation

Recently I am reading Wall's paper "On the Orthogonal Groups of Unimodular Quadratic Forms II". In this paper, I encountered with the map $E^1_\omega$, which now I am interested in. Let $X$ be an ...
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### A question about smooth convex lattice polygons

Let $P$ be a smooth convex lattice polygon in $\mathbb{R}^2$ (the lattice being $\mathbb{Z}^2$). Here smooth means that at any vertex of $P$, the two primitive integer vectors (i.e. vectors whose ...
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### Uniform lattice in semidirect product

A uniform lattice in a locally compact group $G$ is a discrete subgroup $\Gamma\subset G$ such that $G/\Gamma$ is compact. My question is whether a uniform lattice exists in the group  G={\mathbb R}^...
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### What's in the genus of the cubic lattice?

I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...
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### Is there a nice choice-free argument to count the number of sublattices?

It's a well known fact that the number of index $n$ sublattices of a rank two lattice $\Lambda$ is given by $\sigma_1(n) = \sum_{d\mid n} d$. Here is a proof of this fact: Proof: choosing a basis of ...
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### Lattice points near a curve

Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance ...
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### Two types of random walkers on square lattice

Consider a two dimensional square lattice ($n$ by $n$), which is our space $S$ (each point labelled by an index $1\to n^2$), containing two types of particles, distinguished here by either an index $1$...
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### Probability a random matrix contains a short integer vector in its kernel

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...
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### Permutation covering of a $G$-lattice

Let $G$ be a finite group. By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$. We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...
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### Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semi-simple Lie group?

Is the braid group with $n$ strings $\mathcal{B}_n$ known to be a lattice in a connected semi-simple Lie group ? (for $n$, say, bigger than $3$) Or is it known that it cannot be such a lattice ?
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### Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every $x′>x$ and $y′>y$, we have \begin{equation*} f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′). \end{equation*} Suppose $f$ and $g$ are supermodular, non-...
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### Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
### Bounding the number of lattice points inside an $n$-dimensional ellipsoid
I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie inside an $n$-dimensional ellipse. That is, given an $n$-dimensional ellipsoid ...