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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79 views

### Counting lattice points in adelic spaces

Let $\mathbb{A}$ denote the ring of adeles of $\mathbb{Q}$, let $\mu$ be the Haar measure of $\mathbb{A}$, and let $\|\cdot\|_{\infty}$ denote the sup-norm of the components in the Archimedean of $\...

**1**

vote

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29 views

### Defining a notion of “volume of its lattice” for non-rational subspaces

Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”:
$$\...

**5**

votes

**0**answers

145 views

### What kind of locally symmetric space is a rational sphere

Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere.
My question is the following.
Is there other ...

**2**

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96 views

### Can computers take uniform samples from a polytope?

For each $r \in \mathbb N $ write $\mathbb Z/ 10^r = \{a/10^r: a \in \mathbb Z\}$ and $P(r)$ for the lattice $(\mathbb Z/10^r)^N \subset \mathbb R^N$.
Suppose the plane $P \subset \mathbb R^N$ is ...

**2**

votes

**1**answer

173 views

### Is there a relation between the number of lattice points lie within these circles

Suppose we have a circle of radius $r$ centered at the origin $(0,0)$. The number of integer lattice points within the circle, $N$, can be bounded using Gauss circle problem .
Suppose that another ...

**3**

votes

**2**answers

303 views

### Can we count the number of integer lattice points in this case?

Gauss Circle problem gives the number of lattice points lie within a circle of radius $r$. This question points to a reference that estimates the number of lattice points in a $d−$dimensional ball.
$...

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81 views

### How well does convolution model $\mathbb Z$ multiplication?

In $\mathbb K[x]$ or $\mathbb K$ where $\mathbb K$ is a ring we can think of multiplication of polynomials as convolution. Over $\mathbb Z$ this line of thought has led to fast integer multiplication ...

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180 views

### Combinatorial and computational problem related to Weyl groups and the coroot lattice

Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\...

**5**

votes

**1**answer

144 views

### Why is the number of Perfect Matchings in a triangular grid equivalent to the number of Royal Paths?

The sequence A006318 at OEIS stands for the Schröder numbers.
They describes the number of lattice paths from the southwest corner $(0,0)$ of an $n\times n$ grid to the northeast corner $(n,n)$, ...

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votes

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118 views

### Integral subspace generated by a positive semidefinite matrix

Take $\Sigma $ a real positive semidefinite matrix. Define $P$ to be the smallest projection with the property that for any $\mathbf{a}\in \mathbb{Z}^n$ with $\mathbf{a}^\dagger (I-P)^\dagger \Sigma (...

**3**

votes

**1**answer

127 views

### Lattice projections

I imagine the following result is folklore
Theorem. Those $k$-dimensional subspaces $\zeta \subset \mathbb{R}^n$ $(1 \leq k \leq n-1)$ for which the orthogonal projection of the lattice $\mathbb{Z}^n$...

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179 views

### Isomorphism classes of lattices

Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define
$$
V = \{x \in \mathbb R^6 \mid A \cdot x = 0\}
$$
and
$$
\Lambda = \{x \in \mathbb Z^6 \mid A \cdot ...

**4**

votes

**1**answer

171 views

### Problem with the vertices of a convex quadrilateral on integer lattice

I made the following observation and I am wondering if it is always true.
Let $x_1$, $x_2$, $x_3$ and $x_4$ be four positive integer points in the plane ($x_i\in\mathbb{Z^2_{\geq 0}}$) forming a ...

**2**

votes

**1**answer

71 views

### On the laplacian of connected, undirected, multigraphs without loops

Let $G$ be a connected, undirected multigraph, without loops.
Let $L_G = D_G - A_G$,
where $D_G= diag (val (v_1), \ldots , val (v_n) )$ where $n$ is the no. of vertices of $G$ and $val (v_i)$ ...

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41 views

### n-D Gauss circle problem over a rectangle

I would like to approximate the amount of points in $\left(2^{-a\cdot n}\mathbb{Z}^n\right)\cap B^n(0,1)\cap C^n$ where $a>0,\ B^n(0,1)$ is the unit nball and $C^n$ is some rectangular domain ...

**13**

votes

**1**answer

248 views

### Counting primitive lattice points

In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result):
Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then
$$\# ...

**4**

votes

**1**answer

656 views

### Do $G$-invariant non-degenerate quadratic forms come from $G$-invariant even lattices?

The following is a somewhat well-known fact: Given an even lattice $L$ with the pairing $\langle,\rangle: L\times L\to \mathbb{Z}$, we extend the pairing to $L\otimes \mathbb{Q}$ by tensoring with $\...

**0**

votes

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56 views

### Can we solve this using LLL formulation?

Given $N=PQ$ where $P$ and $Q$ are primes and given $T=P+Q$ we can use quadratic equation to solve for factors $P$ and $Q$.
On other hand Coppersmith's class of algorithms use partial information ...

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votes

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41 views

### On fixed dimension integer programming complexity when LLL is perfect

Fixed dimension integer programming (both Lenstra's and Barvinok's) uses LLL which guarantees short vectors but not the shortest possible.
Suppose for a given integer programming problem find $x\in\...

**7**

votes

**1**answer

127 views

### Are there Type III codes with small but nonzero “index”?

Recall that a Type III code of rank $r$ is a linear subspace $C \subset \mathbb F_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $...

**10**

votes

**1**answer

165 views

### Is the “Ramond sector” invariant of a 3-framed lattice always divisible by 24?

For the purposes of this question, a rank-$r$ (integral) lattice is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It ...

**0**

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41 views

### Formalizing a projective additive construction of polytopes?

Given a polytope represented by $$Ax\leq b$$ where $A\in\Bbb Z^{m\times n}$, $x\in\Bbb Z^n$ and $b\in\Bbb Z^m$ with $N\geq 0$ number of integer points can we construct a polytope with $N+1$ integer ...

**11**

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**3**answers

554 views

### Is it possible to completely embed complete Heyting Algebras into upsets of a poset?

Let $H$ be a Heyting algebra. It is a well-known result that there is a partially ordered set (Kripke frame) X such that there is an embedding of Heyting algebras $f: H \to \mathsf{Up}(X)$, where $\...

**7**

votes

**1**answer

167 views

### Shortest vectors in a root lattice

Let $R$ be a simply-laced root system in a Euclidean vector space $E$, with inner product normalized so that every root has length $\sqrt{2}$. Let $L \subseteq E$ be the lattice spanned by $R$. Is ...

**3**

votes

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48 views

### Lattice basis reduction over rings of number fields

Can one use lattice basis reduction algorithms, such as LLL over (low-rank) module lattices over rings of number fields of degree greater than 1? Is there any work on lattice reductions over Euclidean ...

**5**

votes

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96 views

### Averaging number of lattice points in a box over a family of lattices

Consider the diophantine equation
$$ x_1y_1^3 + \dots + x_s y_s^3 = 0. $$
For fixed $\mathbf{y}$ with coprime coordinates this is a $s-1$ dimensional lattice $\Lambda(\mathbf{y})$. Let $N(X)$ denote ...

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votes

**1**answer

318 views

### Maier Phenomena for Gauss Circle Problem

For an arithmetic function $\alpha(n)$, let $S_{\alpha}(x) = \sum_{n \le x} \alpha(n)$. When $\alpha$ is the indicator function of primes, Maier has shown that $$\limsup \frac{S_{\alpha}(x+\Phi(x))-S_{...

**9**

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175 views

### How to count integer lattice points close to a subspace of $\mathbb R^n$?

Consider $m$ linearly independent vectors in $n$-dimensional Euclidean space, $v_1,...,v_m \in \mathbb R^n$ where $1\leq m<n$, and let $U := {\rm span}(v_1,...,v_m)$ denote the $m$-dimensional ...

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183 views

### Sphere packings with antipodal (unequal) spheres

Let $\|\cdot\|_2$ denote the Euclidean norm, let $\langle \cdot, \cdot\rangle$ denote the standard dot product, and let $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...

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59 views

### Siegel's Mean Value Theorem by Macbeath and Rogers

It is claimed in an answer in mathoverflow to a question about Siegel's Mean value theorem (link- Siegel's Mean Value Theorem by Rogers and Macbeath) that there is mistake for the case $n=2$. I ...

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20 views

### A problem about Siegel Mean Value theorem by Rogers and Macbeath

In the paper Siegel's mean value theorem in geometry of numbers (link- https://mathscinet.ams.org/mathscinet-getitem?mr=0103183) Rogers and Macbeath claim in Theorem 1, page 147 that certain sets have ...

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164 views

### Subgroups of $\mathbb{Z}^{n}$ with rotational symmetries

Schmidt (https://projecteuclid.org/euclid.dmj/1077377618) showed that the number of $m$-dimensional subgroups of $\mathbb{Z}^{n}$ of covolume $\leq X$
is
$$c_{1}\left(m,n\right)X^{n}+O\left(X^{n-\...

**1**

vote

**2**answers

529 views

### Examples of groups for which Margulis superrigidity theorem applies

I am not an expert at all in the subject of Lie groups, lattices, arithmetic groups and rigidity. But, lately I am interested in Margulis superrigidity theorem, which in most versions can be stated as ...

**2**

votes

**1**answer

142 views

### Counting lattice points can some give all?

Given convex polytope $\mathcal P\subseteq\Bbb R^n$ with $\mathcal P_\Bbb Z\leq2^n$ integer points and given locations of $O(\log \mathcal P_\Bbb Z)$ integer points in some positions can we obtain $\...

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84 views

### Geometry of a $(d-1)$-dimensional lattice

Let $\mathbf u\in\mathbb Z^d$ be a primitive vector (i.e. $\gcd(u_i)=1$) and let $\Pi_{\mathbf u^\perp}$ be the orthogonal projection perpendicular to $\mathbf u$. I want to understand the geometry of ...

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67 views

### 3D generalization of lattice square grid problem

Consider the following problem: You're given set of lattice points $\{a_i\}_{i=1}^n=\{(x_i,y_i)\}_{i=1}^n$. You have to cover it with lattice square grid having minimum possible number of nodes.
That ...

**4**

votes

**1**answer

128 views

### On necessary condition for no integer points in polytope

For a convex polytope $\mathcal K$ in $\Bbb R^n$ presented by $O(n^c)$ linear inequalities is it true that for $|\mathcal K\cap \Bbb Z^n|=0$ it is necessary that at least one axis of John's ellipsoid ...

**0**

votes

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97 views

### Is there a transformation that scales or shifts number of lattice points?

Unitary transformation preserves volume of a polytope while $SL(n,\Bbb Z)$ preserves lattice points.
Let polytope $\mathcal K$ in $\Bbb R^n$ be presented with polynomial in $n$ linear inequalities.
...

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votes

**3**answers

300 views

### Approximate volume computation and lattice point enumeration - hardness

Both volume computation and lattice point enumeration of convex polyhedron are $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.
...

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**3**answers

342 views

### Checking a generating set of $\mathbb{Z}^k$

Let $a_1, \ldots, a_n \in \mathbb{Z}^k$. I need to check if $a_1, \ldots, a_n$ is a generating set of $\mathbb{Z}^k$, that is, every vector $v \in \mathbb{Z}^k$ can be represented as an integer linear ...

**10**

votes

**3**answers

358 views

### Positive integer combination of non-negative integer vectors

A vector of positive integer numbers with $n$ coordinates is given $a=(a_1,\ldots,a_n)$. It holds that $a_1+\cdots+a_n$ is divisible by some positive integer number $k$. I have checked many cases and ...

**0**

votes

**0**answers

58 views

### On expected intersection of typical lattice with tensor product of cubes?

Consider $T$ tensor product of $t$ origin centered boxes $$\underbrace{[-m_i,m_i]\times\dots\times[-m_i,m_i]}_{n_i\mbox{ times}}$$ each in $\Bbb R^{n_i}$ where $i\in\{1,2,\dots,t\}$ holds.
If we ...

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votes

**0**answers

149 views

### Selmer Group of number fields and Ideal lattices

Let $K$ be a totally real number field of degree $n$ and dicriminant $d$, in this article of F.Lemmermeyer the selmer group of $K$ is defined as
$$\text{Sel}(K):=\{\alpha \in K^{\times}: (\alpha)=...

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125 views

### Formula for Schinzel circle with minimum radius

Schinzel's Theorem states that there are a set of circles with a given number of integer points on the circumference of the circle. The theorem includes the equation for an instance of circle given $n$...

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votes

**1**answer

228 views

### What high dimensional lattices have Voronoi cells that have this property?

Which high-dimensional lattices (particularly $Z_n^*,D_n,D_n^*,A_n,A_n^*$), exhibit the following property shown in the attached diagram? Two 2D lattices are shown, with the lattice points in red, the ...

**2**

votes

**0**answers

90 views

### Gap in Successive minima on lattice spanned by rational and integer combination of integer vectors

We are given a rank $r$ matrix $B\in\Bbb Z^{k\times n}$ where $0\leq r\leq k\leq n$ holds.
We have
$$\mathcal L_\Bbb Z=\{uB\in\Bbb Z^n:u\in\Bbb Z^k\}\subseteq\mathcal L_\Bbb Q=\{uB\in\Bbb Z^n:u\in\...

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votes

**2**answers

90 views

### Integer points spanned by real, rational and integer combination of integer vectors

We are given a rank $r$ matrix $B\in\Bbb Z^{k\times n}$ where $0\leq r\leq k\leq n$ holds.
We have $\mathcal L_\Bbb Z\subseteq \mathcal L_\Bbb Q\subseteq\mathcal L_\Bbb R$ where
$$\mathcal L_\Bbb Z=\...

**5**

votes

**1**answer

125 views

### Bounded version of linear and quadratic Hasse--Minkowski theorem

The Hasse-Minkowski theorem states that if
$$Q(x_1,\ldots,x_n) = \sum_{i,j=1}^n a_{ij} x_ix_j$$
is a quadratic form with $a_{ij} \in \mathbb Z$ and $\det (a_{ij}) \neq 0$, then the equation
$$Q(x_1,\...

**1**

vote

**1**answer

75 views

### Closest vertex in a 3D fcc lattice

The 3D fcc (face-centered-cubic) lattice, which has the same packing ratio as the 3D hexagonal close packed lattice, has the following 12 vectors connecting each vertex with its neighbors:
$(1,-1,0),...

**15**

votes

**1**answer

561 views

### Polynomials leaving invariant the Gaussian integers

It can easily be shown that if a complex polynomial $P$ leaves invariant $\mathbb{Z}$ ($P(\mathbb{Z}) \subseteq \mathbb{Z}$) then it must be a linear combination (with integer coefficients) of Hilbert ...