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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])

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Assume $\Gamma$ is an infinite set composed of formulas (of finite length), and $A$ is a formula (of finite length). For example, $\Gamma=\{x,y\sqcup z,x_1,x_2,x_3,\cdots\}$, $A=(x\sqcap y)\sqcup(x\...
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I am looking up at the book of Yves Myers "Algebraic Numbers and Harmonic Analysis" where, in a serious treatment of Pisot numbers, they used this particular lemma without proof. The ...
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Let $\Gamma$ be a torsion free Zariski-dense discrete subgroup of $\operatorname{SL}_3(\mathbb{R})$. Then one can show that the cohomological dimension of $\Gamma$ is less than or equal to 5. The ...
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Given a prime $p$ Let $$\phi(p):=\sum_{p_i <p} v_{p_i}(p-1) e_i$$ where $e_i$ is the $i$-th standard basis vector, $v_p(n)$ is the valuation of $n$ for the prime $p$ and $p_i$ is the $i$-th prime ...
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There are many great algorithms for enumeration of vectors in a lattice such as Fincke-Pohst-Kannan, extreme pruning etc, not to mention great implementations such as fplll. Let $L$ be high density (...
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I'm reading Wall's paper "On the orthogonal group of quadratic forms II", and I'm stuck at the following statement: "As each lattice is a direct summand of some $X$ such that the ...
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This is a continuation of the previous question Comparison between first minimum of a lattice and a discrete subgroup in function field. Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}...
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We define the successive minima in a lattice $L$ of rank $n$ as in Daniele Micciancio and Shafi Goldwasser, Complexity of lattice problems. A cryptographic perspective, The Kluwer International Series ...
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Let $U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ denote the standard hyperbolic plane. I am trying to construct, for a given natural number $N$, an explicit example of a rank 3 lattice $...
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$\DeclareMathOperator\rank{rank}$I have the following question concerning lattices in the sense of free $\mathbb{Z}$-modules equipped with an integral symmetric bilinear form, possibly indefinite. Let ...
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Let $T=\left[\begin{array}{cc}A&B\\C&D\end{array}\right]$ be an $(m+n)\times(m+n)$ matrix over a finite field ${\mathbb F}_{q}$, where $A$ is $m\times m$ and $D$ is $n\times n$. Consider the ...
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For a number field $K/\mathbb Q$ (as a $\mathbb Q$-vector-space $V$, $n$-dimensional), we have the ring of integers $\mathscr O_K$ (a lattice = copy of $\mathbb Z^n$ in $V$). Ideals $I \subseteq \...
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Let $K$ be an imaginary quadratic number field. If the class number of $K$ is greater than 1 there exist non-principal ideals. Some of the ideals that are principal have prime norm, i.e. they are ...
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Source of considered construction: Morris - Introduction to Arithmetic Groups ; see (6.7.1). Let $F$ be a totally real algebraic number field and fix one real embedding $e: F \to \Bbb R$, such that $...
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I took the last few weeks to read material from other textbooks, but I decided to circle around to Appendix A and build off of my first question. Let $$S(a,m;q) = \frac 1 q \sum_{k = 0}^{q-1} e_q(k^2a ...
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There are many NP-complete problems, e.g. SAT, CVP, SIS, graph colouring, Minesweeper etc. By definition there are polynomial time reductions from one to another of these, at least in their decision ...
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Consider a generalization of the Gauss circle problem and let $$ N(r):= \#\lbrace (x,y,z)\in \Bbb Z^3_{\gt 0}\vert \ln^2(x)+\ln^2(y)+\ln^2(z)\le\ln(r) \rbrace $$ I found that $$N(r)=\left(\frac{2\pi}{...
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Currently, I'm reading the appendix of Bourgain and Rudnick's paper that considers bounds for eigenfunctions of the Laplacian on the flat torus. The proof breaks down for $d > 3$, but in the ...
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For any simple Lie group $\mathrm{G}$ with a finite subgroup $\mathrm{H}$, does there exist a lattice $\Gamma$ with $\mathrm{H}\subseteq\Gamma\subset\mathrm{G}$?
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Let's assume we have a lattice $L \subset {\mathbb R}^n$, given by basis vectors $v^1,v^2,\ldots,v^n$, so $L = \{ \sum_{i=1}^n k_i v^i | k_i \in \mathbb Z \}$. Let's collect all basis vectors into a ...
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I am searching for rational embeddings of positive definite kernels $k$ symmetric, taking rational values and $k(n,n)=1$ such as $k(a,b) = \min(a,b)/\max(a,b)$ or $k(a,b) = 2\gcd(a,b)/(a+b)$ or $k(a,...
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There are several algorithms for lattice reductions in $n$-dimensions, LLL, etc. Here the lattice in question is in ${\mathbb R}^n$ and the basis vectors $b_1, \ldots, b_n$ are usually assumed to be ...
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Let $L$ be a finite graded lattice with rank function $r$. According to the definition on page 39 in the book "Combinatorial Theory" by Aigner, L is said to have the regularity property $(...
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Can every submodular function be represented as a restriction of a graph cut? More precisely, is the following true? Let $f:2^V \to [0,\infty)$ be submodular, symmetric with $f(\emptyset)=0$. Then, ...
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I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of a subring of the hyperreals. It has a non-archimedean ...
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In Theorem 8 of Micciancio’s lecture notes, a reduction from the Closest Vector Problem (CVP) to its optimization version (OptCVP) is given under the assumption that the lattice basis $B \in \mathbb{Z}...
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Let $\Delta_m^n$ be the $n$-dimensional simplex in $\mathbb{R}^n$ with vertices $(0,...,0)$, $(m,0,...,0)$, ... , $(0,...,0,m)$ and let $\tau$ be a lattice triangulation of $\Delta_m$. I am trying to ...
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Let $q$ be a power of a prime $p$, $\Bbb F_q[\theta]$ the analog of $\Bbb Z$ in finite characteristic, $\Bbb R_\infty:=\Bbb F_q((\theta^{-1}))$ the analog of $\Bbb R$ and {$\Bbb C_\infty:=$ the ...
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I understand the LLL algorithm for finding approximate shortest vector in $\mathbb{Z}$-lattices (where the norm function is either $\ell_\infty$ or $\ell_2$), as well as finding the shortest vector in ...
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Let $q$ be a power of a prime $p$, $\Bbb F_q[\theta]$ the analog of $\Bbb Z$ in characteristic $p$, and $\Bbb C_\infty$ the analog of $\Bbb C$ in characteristic $p$. A lattice of rank $r$ in $\Bbb C_\...
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An $n$ dimensional lattice is the set of integer linear combinations of $n$ linearly independent vectors in $\mathbb{R}^{d}$ ($d\le n$). The $n$ independent vectors are called the basis of the lattice,...
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I've found in the literature these facts: Any closed flat manifold is virtually (i.e. finitely covered by) a torus, and any finite-volume real hyperbolic manifold has virtually (i.e. is finitely ...
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This is the continuation of the previous question Lattices in the extension field of local fields in positive characteristic. Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}_q$, where ...
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I believe his thesis constructs a 248-dimensional representation of the Dempwolff group and is related to the construction of the rank 248, even self-dual Smith-Thompson lattice. I have tried the ...
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Here is the 2-dimensional integer lattice reachability problem: Suppose we are in a 2d integer plane, and are starting from $(0, 0)$. We will add $n$ vectors, and each vector is from a pre-defined ...
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An $n$-dimensional Euclidean lattice $L$ is called of $\textbf{Voronoi's first kind}$ if it satisfies the $\textbf{obtuse superbasis}$ condition: There exist $x_0, \dots, x_n \in L$ such that $1) x_1, ...
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Let $A$ be a non-singular matrix in $\mathbb{R}^{n \times n}$. Let $S^{n-1}$ denote the surface of the unit $n$-sphere in $\mathbb{R}^{n}$. Suppose we know that there exists an $x \in S^{n-1}$ such ...
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Can you show an example of a lattice and its dual where the product of the length of their smallest vectors is close to the dimension of the lattice? Let $L$ be an $n$-dimensional lattice, and let $L^{...
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Is there a known lower bound on the degree $D$ of a Real polynomial $P$ of $d$ variables which is non-zero yet vanishes on all the lattice points in the closed ball of radius $r$ about the origin? How ...
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Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}_q$, where $q$ is a prime power. The norm in this field is defined by $ \left| \frac{f}{g} \right| = q^{\deg(f) - \deg(g)}, $ where $f, ...
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There seems to be some folklore result that the number of lattice points in a $d$-dimensional ball of radius $R$ is $$V_d R^d + \mathcal O(R^{d-2})$$ and $V_d$ is the volume of the unit ball if $d$ ...
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We consider the space of $n$-dimensional Euclidean lattices. Define the $i$-th successive minimum of a lattice $\Lambda$ as $ \lambda_i(L) := \min \left\{ r \in \mathbb{R}_{>0} \mid \dim_{\mathbb{R}...
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Let $P$ be the positive integer point lattice on the plane, that is, all points $(x,y)\in\mathbb{R}^2$ such that $x,y\in\mathbb{N}, x,y>0$. Take $a_i,b_i,c_i\in P$ such that $b_i=(b_{i,x},b_{i,y}) \...
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Considering a quadratic congruence equation $$ x^2 = d \pmod{p} \label{1}\tag{1} $$ I have a strange idea: what if we construct a system of congruence equations in multiple variables based on the ...
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Given a matrix $A\in M_{6\times 6}(\mathbb{Z})$ that is symmetric and has determinant zero. I want to deterministically figure out if there exists a matrix $T\in M_{6\times 5}(\mathbb{Z})$ such that $...
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A duplicate of this: I am reading Mahler’s Lattice points in $n$-dimensional star bodies II (1946), in which he poses the following (then) open problems about the irreducibility of star bodies: Do ...
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$LLL$ algorithm is vectorized version of Extended Euclidean algorithm for $\mathsf{GCD}$. Even the $m=2$ dimensions case known to Lagrange and Gauss does not have an $NC$ algorithm for shortest vector....
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Conjecture: $$\forall \Lambda,\ \exists P(x)\in \mathbb{Z}[x],\ S(\Lambda):=\sum_{k\in\Lambda}\prod_{j=1}^{n}\frac{1}{1+k_{j}^2}=\frac{\pi^{n}}{\sinh^{n}(\pi)\operatorname{d}(\Lambda)}P\left(\cosh\...
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In "Symplectization, Complexification and Mathematical Trinities" by V. Arnold, he came up with various trinities, one of which being the three Platonic symmetry groups $T,O,I$ of order $12, ...
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Let $H\subset \operatorname{GL}(n,\mathbb{C})$ be a connected, semisimple algebraic group defined over $\mathbb{Q}$. Fix a number field $K$ with $[K:\mathbb{Q}]=3$ that is not totally real. Denote its ...
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