Questions tagged [lattices]
Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])
652 questions
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Rational linear independence of holomorphic functions
Fix an integer $ m $ and a lattice $ \Lambda \subset \mathbb{C}^{m} $. Identify $ \Lambda \otimes \mathbb{R} $ with $ \mathbb{C}^{m} $.
Take $ n $ holomorphic functions $ f_{1}, \ldots, f_{n}: U \to \...
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Which even lattices have a theta series with this property?
This is a slight generalization of a question I made in Math StackExchange, which is still unanswered after a month, so I decided to post it here. I am sorry in advance if it is inappropriate for this ...
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Lattices are not solvable in non-compact semisimple Lie groups
I'm trying to prove the following result.
If $G$ is a non compact semisimple Lie group with no compact factors (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is ...
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Find a lattice basis given too many points
Fix a discrete addition subgroup in $\mathbb{R}^n$. Given a finite spanning set, how can one find a group basis?
3
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answer
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Convex Hulls of Demazure Modules
Let $G$ be a semisimple algebraic group over $\mathbb{C}$ and for a highest weight $\lambda$, denote by $V_{\lambda}^w$ the Demazure module associated with $\lambda$ and $w$. More precisely, $V_{\...
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Selfsimilar lattices in $\mathbb R^d$
Let $\Lambda\subset \mathbb R^d$ a discrete subgroup, up to diminishing $d$ we assume it is of the form $A\mathbb Z^d$ with $A\in GL(d)$. Up to dilation we assume that the shortest vector in $\Lambda\...
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Relating the shortest vector of a lattice to the orthogonal complement of the lattice
By a lattice we mean sub-lattice of $\mathbb{Z}^n \cap V$, where $V$ is a subspace of $\mathbb{R}^n$ defined over $\mathbb{Q}$. We say that a lattice $\Lambda$ is primitive if a basis of $\Lambda$ can ...
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Partitioning $\{0,1\}^n$ into $n$ sets
I am working on an answer to the question
Magic trick based on deep mathematics
and came across the following problem: I am trying to partition the cube $\{0,1\}^n$ into $n$ sets $P_1,\dots,P_n$ ...
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Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients
Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above:
$$(i)\ 1 + 80q^3 ...
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Constructive proof of Swan theorem
Let $M$ be an $S_n$-lattice (so it is free as an abelian group), and assume that $M$ is projective (i.e. direct summand of some $\mathbb Z[S_n]^m$). A theorem of Swan implies that $M$ is stably ...
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List of Automorphism groups of Abelian Varieties for Dummies
(%Edited after abx comment%)
I seek explicit linear integral representations $\rho: Aut(X,\omega) \to Sp_{2g}(\mathbb{Z})$, when $(X,\omega)$ is complex $g$-dimensional PPAV. I prefer explicit ...
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An interesting sum over lattice points in a large disk centered at the origin
Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits_{(m,n) \in D_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^2 + n^2$, where $D_r$ denotes the closed ...
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Smallest integer lattice point by box measure in a polytope?
Given an integer lattice $\mathcal L\subseteq\mathbb Z^n$ represented by basis $\mathcal B$ and an integer linear program $Ax\leq b$ where $x\in\mathbb Z^n$ is unknown with $A\in\mathbb Z^{m\times n}$ ...
- 13.9k
1
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1
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Lowering $i$th shortest vector of a lattice
LLL guarantees that we can find a basis $v_1,\dots,v_n$ of a lattice in $\mathbb R^n$ with
$$\|v_i\|\leq \gamma_{i,n} \det(\Lambda)^{1/(n-i+1)}$$ where $\gamma_i$ is a function only of $i$ and $n$.
...
- 13.9k
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Divisor bound for $r_2$ off the origin
If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
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Which lattices are rotatable into their scaled copy?
Let $L=\{\sum_i n_iv_i\mid n_i\in\mathbb Z\}$ be some lattice generated by $d$ independent vectors $(v_i)_1^d$ from $\mathbb R^d$. Call $L$ rotatable if for some $M$, a scalar multiple of some ...
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Shortest Lattice Vector with restricted $x$
Let $\Lambda$ be a lattice with basis, $B$ consisting of vectors $b_i$, so that the elements of $\Lambda$ are of form, $y\in \Lambda \iff y=Bx=\sum_i b_ix_i$ for some $x_i\in\mathbb{Z}$.
My questions ...
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Basis of cone lattice
I only want to know whether a construction that I use appears in literature and maybe has a name already.
Let $V$ be a $\mathbb Q$ vector space of dimension $d\in\mathbb N$.
A subset $C\subset V$ is ...
user130903
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Coppersmith's method to quadrivariate degree $2$ polynomials that behave as bivariate?
We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\...
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Method of Coppersmith optimal for multivariate?
It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials ...
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Given an integer lattice, how to count the number of points whose norm is smaller than some bound $M$?
Let $\mathbf{b}_1, \mathbf{b}_2, ..., \mathbf{b}_n$ be linearly independent $m$-dimensional vectors whose entries belong to $[0, M] \cap \mathbb{Z}$, for some $M \in \mathbb{N}^*$. Of course, $n \le ...
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Counting lattice points in adelic spaces
Let $\mathbb{A}$ denote the ring of adeles of $\mathbb{Q}$, let $\mu$ be the Haar measure of $\mathbb{A}$, and let $\|\cdot\|_{\infty}$ denote the sup-norm of the components in the Archimedean ...
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Defining a notion of “volume of its lattice” for non-rational subspaces
Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”:
$$\...
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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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Shortest vectors in tensor product and maximal lattices in tensor product
$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$
$$\|v_1'\|_2\...
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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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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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Lattice points in a rotated product-of-balls
Fix $U$ unitary over $\mathbb{R}^{K},$ take $U_n=I_{n\times n}\otimes U$ and denote the unit ball at 0 in $\mathbb{R}^n$ as $B^n$. For $d_1,\dots,d_K>0$, fix $S_n:=U_n\left(\prod_{k=1}^K d_k B^n\...
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1
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The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)
Original question (without additional information from Wendy):
Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way:
Taking the E8 as {128,...
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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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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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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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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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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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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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1
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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 $val (v_i)$ ...
user111492
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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
$$\# ...
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Do $G$-invariant non-degenerate quadratic forms come from $G$-invariant even lattices?
The following is a somewhat well-known fact: Given an even lattice $L$ with the pairing $\langle,\rangle: L\times L\to \mathbb{Z}$, we extend the pairing to $L\otimes \mathbb{Q}$ by tensoring with $\...
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Are there Type III codes with small but nonzero "index"?
Recall that a Type III code of rank $r$ is a linear subspace $C \subset \mathbb F_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $...
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Is the "Ramond sector" invariant of a 3-framed lattice always divisible by 24?
For the purposes of this question, a rank-$r$ (integral) lattice is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It ...
- 54.6k
8
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Shortest vectors in a root lattice
Let $R$ be a simply-laced root system in a Euclidean vector space $E$, with inner product normalized so that every root has length $\sqrt{2}$. Let $L \subseteq E$ be the lattice spanned by $R$. Is ...
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Lattice basis reduction over rings of number fields
Can one use lattice basis reduction algorithms, such as LLL over (low-rank) module lattices over rings of number fields of degree greater than 1? Is there any work on lattice reductions over Euclidean ...
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Averaging number of lattice points in a box over a family of lattices
Consider the diophantine equation
$$ x_1y_1^3 + \dots + x_s y_s^3 = 0. $$
For fixed $\mathbf{y}$ with coprime coordinates this is a $s-1$ dimensional lattice $\Lambda(\mathbf{y})$. Let $N(X)$ denote ...
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1
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Maier Phenomena for Gauss Circle Problem
For an arithmetic function $\alpha(n)$, let $S_{\alpha}(x) = \sum_{n \le x} \alpha(n)$. When $\alpha$ is the indicator function of primes, Maier has shown that $$\limsup \frac{S_{\alpha}(x+\Phi(x))-S_{...
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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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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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2
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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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1
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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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