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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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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 $\...
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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”: $$\...
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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 ...
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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 ...
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1answer
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 ...
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2answers
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\...
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1answer
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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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 (...
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1answer
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 ...
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1answer
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 ...
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1answer
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 ...
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1answer
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 $$\# ...
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1answer
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 $\...
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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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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\...
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1answer
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 $...
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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 ...
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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 ...
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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 $\...
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1answer
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 ...
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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 ...
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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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1answer
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_{...
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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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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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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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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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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-\...
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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 ...
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1answer
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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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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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 ...
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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 ...
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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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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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3answers
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 ...
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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 ...
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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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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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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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1answer
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 ...
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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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2answers
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=\...
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1answer
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,\...
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1answer
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),...
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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 ...