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])
231
questions with no upvoted or accepted answers
2
votes
0
answers
202
views
Modular inverse computation - avoiding Euclidean algorithm
Modular inverse is known to be computable by Extended Euclidean algorithm which is the reaping the rewards of computing the GCD of two numbers or proving two numbers are coprime.
If we already know ...
2
votes
0
answers
124
views
Non-uniform lattices and parabolic subgroups in Lie groups
Let $G$ be a semisimple connected Lie group and let $\Lambda < G$ be a non-uniform irreducible lattice.
How does it follows that there exists a minimal parabolic subgroup $P$ of $G$ such that the ...
2
votes
0
answers
85
views
To show $\{(x,y) \in \mathbb Q^{\geq 0} \times \mathbb Q^{\geq 0}~:~ mn+1 \mid m^x+n^y \}$ is subset of the lattice $\{\vec u+i \vec v+j \vec w\}$?
I am writing two definitions, the $1$st one is a cover in some sense while the $2$nd one is a lattice:
Definition 1: If $m,n$ are integers bigger than $1$, then the set $$A=\{(x,y) \in \mathbb Q^{\geq ...
2
votes
0
answers
170
views
Mistake in Rogers' paper: "number of lattice points in a set" for the case $n=2$?
Let $f:\mathbb R^n\to \mathbb R$ be a nonnegative Borel measurable function, and
let $f^*$ be the function obtained from $f$ by spherical symmetrization (see Rogers' paper: number of lattice points in ...
2
votes
0
answers
132
views
Lattice relations and isogenous elliptic curves
Consider two (primitive) elements $\pi_{i} \in \mathbb{C}$, such that $\pi_{1} = M \pi_{2}$ for $M \in \mathcal{S}_{m}$ with $$\mathcal{S}_{m}:=\Big\{\begin{pmatrix}
A & B \\
0 & D
\end{...
2
votes
0
answers
105
views
Find the closest point of a lattice $\Lambda$ given the closest point for its union of cosets $\bigcup_i ({\bf r}_i+\Lambda)$
Suppose we have an $n$-dimensional lattice $\Lambda$, and a set of vectors ${\bf r}_i$, we can construct a union of cosets of $\Lambda$, denoted as $L$, as
$$
L \equiv \bigcup_i ({\bf r}_i+\Lambda)
$$
...
2
votes
0
answers
128
views
Bruhat-Tits tree as Cayley graph of free group
$\DeclareMathOperator\BT{BT}\DeclareMathOperator\GL{GL}$Let $p > 2$ be a prime and $n = \frac{p + 1}{2}$. We can identify the vertices of Bruhat-Tits tree $\BT(\mathbb Q_p)$ with the elements in ...
2
votes
0
answers
100
views
Kac-Peterson modular forms and shifted theta functions
Let $\Lambda$ be the root lattice corresponding to an ADE root system $R$ of rank $n$. With the ADE assumption, the weight lattice is simply the dual lattice $\Lambda^{\vee}$. Given any weight vector $...
2
votes
0
answers
87
views
Is there an exact solution for the number of points within a circle of radius r for an honeycomb lattice?
I want to ask if exists an exact solution for the number of points within a circle of radius r for an honeycomb lattice.
I know that it is exist for an square lattice https://mathworld.wolfram.com/...
2
votes
0
answers
55
views
Does a transitive action of $O(L^{\#}/L)$ imply a transitive action of $O(L)$ on $L^{\#}/L$?
Given an even lattice $L$ the orthogonal group $O(L)$ acts on the discriminant group $L^{\#}/L$ as is known. Hence, for every $\gamma \in O(L)$ there is a $\overline{\gamma}\in O(L^{\#}/L)$ that acts ...
2
votes
0
answers
80
views
Showing an action of a higher rank lattice on hyperbolic space has a fixed point
In the introduction to this paper, the author mentions that any action of a lattice $\Gamma < G$ on a rank one symmetric space $X$ has a fixed point, where $G$ is a higher rank semisimple algebraic ...
2
votes
0
answers
133
views
density of lattices
I'm looking for references pertaining to the remark at the bottom of p.18 of Conway-Sloane, "Sphere Packings, Lattices and Groups" (3rd ed), henceforth referred to as "SPLG".
First,...
2
votes
0
answers
104
views
Computer program which computes the automorphism group of Gram Matrix of lattice?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Fixed $n \geq 2$, given $K \in \GL(n,Z)$. One can view $K$ is a Gram matrix of Lattice. I also imposed that $K$ is symmetric i.e $K^{T}=K$. We ...
2
votes
0
answers
90
views
Sublattices in the standard integral symplectic lattice
Let $V$ denote $\mathbb{Z}^{2g}$ with its standard integral symplectic form $\omega = \sum_{i=0}^{g-1}dx_{2i} \wedge dx_{2i+1}$ (or, the homology lattice of a genus $g$ surface with its intersection ...
2
votes
0
answers
115
views
How much Gleason type theorem do I need? Quasi states vs. states
Let $\varphi$ be a quasi state on $B(H)$. What does it mean? It means that $\varphi(cA)=c\varphi(A)$ for $c \in \mathbb{C}, A \in B(H)$, $\varphi(A) \geq 0$ for positive $A$ and $\varphi(A+B)=\varphi(...
2
votes
0
answers
128
views
A number theoretic unlikely intersection phenomenon
Let $h_1, \cdots, h_k$ be co-prime integers, $k \geq 3$. Let $n$ be a square-free positive integer, understood to be extremely large compared to the $h_i$'s, and such that $\omega(n)$, the number of ...
2
votes
0
answers
123
views
Another generalization of the Gauss circle problem
In this question I asked for a generalization of the Gauss circle problem, the type of generalization I am asking is to view the Gauss circle problem as one about counting algebraic integers of ...
2
votes
0
answers
120
views
Distribution of Smith normal forms for lower triangular matrices with given diagonals
Given integers $m$ and $n$ and $d_1, \ldots, d_m \in \mathbb{Z}/n \mathbb{Z}$, consider the set of all lower-triangular matrices of dimension $m$ with diagonal elements equal to $d_i$. What can be ...
2
votes
0
answers
43
views
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 \...
2
votes
0
answers
122
views
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\...
2
votes
0
answers
85
views
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 $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$
$$\|v_1'\|_2\...
2
votes
0
answers
110
views
Can computers take uniform samples from a polytope?
For each $r \in \mathbb N $ write $\mathbb Z/ 10^r = \{a/10^r: a \in \mathbb Z\}$ and $P(r)$ for the lattice $(\mathbb Z/10^r)^N \subset \mathbb R^N$.
Suppose the plane $P \subset \mathbb R^N$ is ...
2
votes
0
answers
209
views
Combinatorial and computational problem related to Weyl groups and the coroot lattice
Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\...
2
votes
0
answers
118
views
Gap in Successive minima on lattice spanned by rational and integer combination of integer vectors
We are given a rank $r$ matrix $B\in\Bbb Z^{k\times n}$ where $0\leq r\leq k\leq n$ holds.
We have
$$\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\...
2
votes
0
answers
175
views
Lattice Sieving in Number Field Sieve
I am currently going through Pollard's article on Lattice Sieving and have a few confusions. Firstly, how to figure that $C$ and $D$ in the two-dimensional array so that every $(c,d)$ pair corresponds ...
2
votes
0
answers
51
views
Bound on the fundamental region of a sublattice of a Niemeier Lattice
I am working with an 18 dimensional lattice, $W$ say, primitively embedded into a given Niemeier lattice $\mathcal{N}_i$. I am trying to figure out the following: what is the smallest possible ...
2
votes
0
answers
87
views
Minimal diameter of a class in a number field
Let $\mathbb{K}$ be a number field of degree $n$, let $A = \mathcal{O}(\mathbb{K})$ be the ring of integers and consider the Minkowski embedding
$\mathbb{K} \rightarrow \mathbb{R}^n$ given by $x \...
2
votes
0
answers
89
views
On design of a (preferrably unimodular) matrix
Assume each entry is in $\Bbb Z$.
Say we want to solve $Ax=b$ where known $A$ is $n\times n$, unknown $x$ is $n\times1$ and $b$ is $n\times1$.
The absolute value of minors of augmented matrix $[A|b]$...
2
votes
0
answers
219
views
Integrating an n-fold Cauchy product of a Fourier series
I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found here.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be ...
2
votes
0
answers
121
views
Hamming weights of special vectors
The motivation of this question comes from number theory (I add the tag number theory for this reason, in that it is possible that someone with a number-theoretic background has already thought about ...
2
votes
0
answers
77
views
automorphic forms associated with symmetries of vertices of uniform honeycombs in hyperbolic space
Is there a catalogue of automorphic forms (modular/Maass/Siegel/Hilbert...) which lists them in terms of Poincaré series associated with the symmetries of the vertices of uniform honeycombs in ...
2
votes
0
answers
450
views
Finding good high-dimensional sphere coverings in Euclidean space
Suppose we want to cover the unit sphere $\mathcal{S}^{d-1} := \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ with spherical caps $\mathcal{C}_{\mathbf{y}} := \{\mathbf{x} \in \mathcal{S}^{d-1}...
2
votes
0
answers
62
views
Dual equivalence for multioperators
This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...
2
votes
0
answers
132
views
$\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$
I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?
2
votes
0
answers
286
views
Is the kissing number in $n$ dimensions always divisible by $n$? And what is the base of exponential growth of the kissing number?
And why are the kissing numbers for 1, 2, 3, 4 and 8 dimensions all highly composite numbers?
2
votes
0
answers
159
views
Dislocations,Disclinations Latices, Displacement fields and scaling
I am looking up Frank, and Burger vectors and associated material on dislocation/disclination. It seems straightforward describing a lattice and what dislocation means. It is even possible to restrict ...
2
votes
0
answers
123
views
Are Zariski-dense representations of a cocompact complex hyperbolic lattice non-obstructed?
Question
Suppose that $\Gamma < \text{SU}(n,1)$ is a cocompact lattice, and let $\rho \colon \Gamma \to G$ be a representation to a non-compact simple Lie group (most interesting case for me: $G = \...
2
votes
0
answers
214
views
Standard name for a Monoid/Semigroup with $a+b \leq a, b$?
I have seen suplattice and inflattice being used when dealing with a lattice. What about when you don't have a lattice?
For instance, for reals $a,b > 0$, define $$a \oplus b = \frac{1}{\frac{1}{a}...
2
votes
0
answers
196
views
Existence of inclusions of finite groups with a particular lattice property
Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by :
$(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a finite group morphism and ...
2
votes
0
answers
88
views
Group actions on polytopes in indefinite integer lattices
Is anything at all known about polytopes in indefinite integer lattices? I'm interested in lattice automorphisms which preserve certain polytopes of "high regularity" (e.g. cones). As a first step, I'...
2
votes
0
answers
128
views
Reference request for gluing construction of lattices
I would like to study gluing method of lattices (such as constructing Niemeier lattices from certain root lattices etc) and am looking for good references. I am aware of the book "Sphere Packings, ...
2
votes
0
answers
115
views
Finite subgroups (lattices) in the large N limit of SU(N)
I would like to gain some information about the discrete subgroups (lattices) of SU(N) Lie groups. I have already read some answers and references concerning the N=3 and N=4 cases. I am more ...
2
votes
0
answers
243
views
Hurwitz integers and $F_4$
The Hurwitz integers are
$$
\mathcal H=
\{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}.
$$
I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...
2
votes
0
answers
343
views
Constructing the Stone Space of a Distributive Lattice
Does anyone have a good reference for the method of giving a topology to a distributive lattice as outlined in M.H. Stone's "Topological representation of distributive lattices and Brouwerian logics"? ...
2
votes
0
answers
246
views
Simultaneous diophantine approximation with polynomial bound
For a given number $\alpha$ continued fractions expansion $(p_n, q_n)$ of $\alpha$ has the remarkable property that not only $|\alpha - \frac{p_n}{q_n}| < \frac{1}{q_n^2}$, but the converse holds - ...
2
votes
0
answers
163
views
Minimally 6-connected 3D discrete lines that are convex lattice sets
There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that ...
2
votes
0
answers
330
views
Upper bound on the number of shortest vectors in a lattice
Given a Lattice $L$ (e.g. by its Gram-Matrix or via a basis) I would like to know whether there is an upper bound on the number of shortest vectors in $L$. Available information on $L$ includes ...
1
vote
0
answers
96
views
Number of points in a ball in positive characteristic
Let $w_1,\cdots,w_n$ be of elements of $\Omega$, that is the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by the $-\deg$ valuation.
Assume that $w_1,...
1
vote
0
answers
55
views
Condition on the minimality of Minkowski units
I am interested in to undrestand when the Minkowski units in real biquadratic number fields are minimal in the log unit lattices.
I have read some pieces of literature online which are investigating ...
1
vote
0
answers
29
views
Does $2$ variable linear Diophantine equation in $NC$ imply $2$ dimensional shortest vector is in $NC$?
Consider the Linear Diophantine in known $a,b,c\in\mathbb Z$
$$ax+by=c.$$
Above can be solve by Extended Euclidean which is not in $NC$ as far as we know. It is clear if Extended Euclidean is in $NC$ ...