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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25
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502 views

Is there an Ehrhart polynomial for Gaussian integers

Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
20
votes
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797 views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
12
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0answers
231 views

Who first showed that $SL(n,O_K)$ is a lattice for a number ring $O_K$?

Let $O_K$ be the ring of integers in an algebraic number field $K$. Assume that $K$ has $r$ real embeddings and $s$ pairs of complex conjugate complex embeddings. There is then an injective ...
12
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492 views

Connes & Marcolli: Q-lattices generalize Conway's “Understanding groups like $\Gamma_0(N)$”

Has anyone generalized Conway's description of Hecke operators on lattices to the Q-lattices of Connes & Marcolli ? Light may well be shone on moonshine thus.
8
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272 views

What are the homological properties of Young's lattice?

Young's lattice $Y$ is a graded poset and a distributive lattice whose elements are all the partitions of $n$ for $n \in \mathbb{N}$ with the poset relation coming from inclusion of Young diagrams. ...
7
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237 views

Question on some coverings of the euclidean space

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has ...
7
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189 views

Minkowski's convex body theorem for ellipsoids

Minkowski's theorem states that if $K\subseteq\mathbb{Z}^n$ is a convex compact set, $K=-K$, and $\mathrm{volume}(K)\geq 2^n$, then $K$ contains a nonzero integral vector. Can this bound be improved ...
6
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151 views

Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying $b$ is bilinear, $b(...
6
votes
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343 views

Is this obfuscation scheme unbreakable?

I've just come across this popular article about a breakthrough (which can be purchased here), published in Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium by a team of ...
6
votes
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487 views

What is the structure of a space of $\sigma$-algebras?

Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
5
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59 views

Almost invariance in compact quotients of locally compact groups

While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups in the topological setting, I face the following problem. Let $G$ be a locally compact amenable ...
5
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216 views

Bound on the number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...
5
votes
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91 views

Minimum of the product of linear forms over a lattice

In Chapter [IX.1] of Siegel's Lectures on the Geometry of Numbers it is shown that if we have $n$ linear forms $y_{j}=\sum_{k=1}^{n}{a_{jk}x_{k}},\quad j=1,\ldots,n$, with the coefficient matrix $(a_{...
5
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0answers
235 views

Lattice points inside a (n-dimensional) tetrahedron

Hi, overflowers. I was interested in a sharp lower bound for the number of lattice points (say, integral lattice points) inside the tetrahedron defined by the coordinate hyperplanes and $x_1/a_1+...+...
5
votes
0answers
836 views

On “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)

I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" ...
5
votes
0answers
171 views

Effect of Covering Radius on Shortest Vector

For "even" integral lattices in dimension at least 4, does a covering radius strictly less than $\sqrt 2$ imply that there is a vector of norm 2, also called a root? Note that this is simply false in ...
5
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376 views

Have you seen this sort of an anti-involution on a lattice?

While looking at a representation theory question, I came up with the following sort of object. I want to know if it comes up often in combinatorics or some other area of mathematics. Let $P$ be a ...
4
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40 views

Deformations of null-vectors of an integral unimodular lattice

Is there an $SO(n,n)$ transformation that makes the Euclidean norm-squared of all the null vectors of the $(n,n)$ hypercubic lattice strictly greater than 4? Example: Let $\Lambda_{\rm sL}$ denote ...
4
votes
0answers
283 views

Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there. Recall that a subfactor is Dedekind if all its intermediate subfactors are normal. A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
4
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132 views

Commutative quantale (Q,+) with a binary operation distributing over +

Context: In Flagg's "Quantales and continuity spaces" the notion of a continuity spaces enriched in a certain quantale is introduced. Abstracting from the set $[0,\infty]$ the essential properties ...
4
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226 views

Domination in Nice Lattices

Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions. Call a lattice ...
3
votes
0answers
57 views

Lattice points in regular simplex

Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to ...
3
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139 views

Lenstra's integer programming algorithm: Finding a lattice point “near the center”

I have already posted this question on the mathematics forum, but I suspect the question needs more detailed knowledge than most users have; please excuse the duplicate post. Any help is greatly ...
3
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0answers
275 views

What's the ratio of inclusions of finite groups with a distributive lattice?

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$. ...
3
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0answers
120 views

Lattice with trivial spinor norm

Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows: For a ...
3
votes
0answers
143 views

references for properties/examples of breadth in (semi)lattices

This is in some sense following up on my earlier question and the answer given by NN. I am currently revising the paper which used the condition mentioned in my question. It was pointed out in NN's ...
3
votes
0answers
184 views

Automorphisms of Torsion Quadratic Forms

Let $L$ be an $n$-dimensional lattice with an integer valued quadratic form $q$. Fix a basis $e_i$ for $L$ and let $K_{ij} = \langle e_i, e_j \rangle$, where $\langle x,y \rangle = q(x+y) - q(x) - q(...
3
votes
0answers
91 views

Versions of Helly's Theorem for Unbounded Parallelpipeds

I'm studying Frobenius splitting of toric varieties based on Sam Payne's characterization of Frobenius splitting ("Frobenius splittings of toric varieties" (2008)) and I came across a sort of Helly's ...
3
votes
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396 views

a poset with small “cycles”

(a followup to this recent question) I noticed the following curious property of a poset (which I strongly believe to be a lattice, I'm still trying to prove that...): Suppose that $z$ is covered by ...
2
votes
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48 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
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0answers
118 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
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147 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
0answers
74 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
0answers
177 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
0answers
184 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
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0answers
260 views

A question on the theorem of Minkowski-Hlawka

The Minkowski-Hlawka theorem (as stated by Gruber in his lovely book Convex and Discrete Geometry) says that if $S$ is a Jordan measurable set in $\mathbb{R}^n$ with volume < 1. Then there is a ...
2
votes
0answers
336 views

Singular fibers of an elliptic fibered K3 surface.

Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...
2
votes
0answers
102 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
0answers
218 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
0answers
305 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
0answers
186 views

bounds for the covering radius (or diameter of the Voronoi cell) of a lattice coming from a number field

Let $K$ be a number field and $O$ an order in $K$. Denote the real embeddings of $K$ by $\sigma_1,\dots,\sigma_s$ and the non-real complex embeddings of $K$ by $\sigma_{s+1},\overline{\sigma_{s+1}},\...
2
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217 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
0answers
149 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
0answers
273 views

Pointwise bounds for Dirichlet kernel over truncated lattice

In 1 dimension, a "one-sided" Dirichlet kernel $D_N(x)=\sum_{k=0}^{k=N}e^{\frac{2\pi}{N}ikx}$ has its module sharply peaked around points corresponding, roughly, to the "dual lattice" $N\cdot\mathbb{Z}...
2
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0answers
143 views

Can lattice join be an element of residuated pair?

Given a classic (not Residuated) lattice, with standard definition of partial order via lattice join and meet operations, is it possible to satisfy Galois equivalence ...
2
votes
0answers
276 views

Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...
1
vote
0answers
68 views

The Linnik problem for dimension $2$

For $N$ an integer, let $$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$ For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...
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0answers
47 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 $$p : L \to Ft(M)^{op}$$ ...
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77 views

Relation to Ehrhart polynomial with Uniqueness

A set of relative prime, positive integers $A = [a_1, \dots, a_d]$ describe the restricted partition function $$ p_A(n) = \# \{(m_1,\dots,m_d)\in\mathbb{Z}^d: \textrm{ all }m_j \geq 0, \sum_{j=1}^d ...