# Questions tagged [integer-lattices]

A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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### Distribution of "good" and "bad" basis in lattice families?

I'm trying to learn more about lattice based cryptosystems. One of the fundamental ideas behind lattice based cryptosystems is that there can be many equivalent basis for a single lattice. Formally, ...
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### Lattice with only isotropic vectors (does not exist)

Consider a lattice $L:=(X, \phi)$, i.e. $X$ is a free $\mathbb Z$ module of finite rank and $\phi:X\times X\to \mathbb Z$ is a non-degenerate, symmetric, bilinear form. An element $x \in X$ is said ...
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### Show that a vector satisfies some properties after basis change

This is the exercise 12-5 from Introduction to Riemannian manifolds of John Lee. Let $(v,w)$ be a basis of $\mathbb{R}^2$ and let $\Lambda$ be the lattice generated by them. Let $\tilde v$ be the ...
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### How "unorthogonal" can a LLL-reduced basis be?

I have been recently studying LLL-reduction. I get from the size condition and Lovasz condition that the basis are guaranteed to be somewhat orthogonal. But I couldn't figure out how orthogonal the ...
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### How far can two points be in the fundamental parallelepiped of a LLL-reduced basis?

Given the lengths $\{d_1, \dots, d_n\}$ of a LLL-reduced lattice basis $\{\mathbf{d}_1, \dots, \mathbf{d}_n\}$, how far can two points be in the fundamental parallelepiped of a LLL-reduced basis? I.e.,...
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### For polytopes in $\mathbb{R}^n$, bound number of lattice points by its volume?

Suppose we are considering polytopes in $\mathbb{R}^n$ given by equations of the form $Ax \le b$. Given that the polytopes are bounded, can we upper bound the number of lattice points of the polytope ...
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### What lattices beyond the laminated lattices (particularly in $\le 24D$) belong to a slightly expanded category that includes "descendants" of Λ13_mid?

Back in 2016, in his proposed answer to the question asked at Packing of n-balls , achille hui ( https://math.stackexchange.com/users/59379/achille-hui ) gave the following definition of laminated ...
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### Finding a lattice vector in $\mathcal{L}(b_1, \ldots, b_n)$ within a distance $(\|b_1\|^2 + \cdots + \|b_n\|^2)^{1/2}/2$ from a given point

Given a lattice $\mathcal{L} \subset \mathbb{R}^m$ with a $\delta$-LLL reduced basis $\{b_1, b_2, \ldots, b_n\}$ and a vector $x \in \mathbb{R}^m$, how do I find a lattice vector $\ell \in \mathcal{L}$...
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I will note $\mathbb{Z}_q = \mathbb{Z}/q\mathbb{Z}$. I read in a lecture of Micciancio about lattices (exercice 5), that for any $q, m, n \in \mathbb{Z}$, we have $\mathbb{P}_{A \in \mathbb{Z}_q^{n \... 1 vote 1 answer 54 views ### If$B$is an integer matrix, does$\mathrm{det}(B^TB)$has to be a perfect square? Suppose$B \in \mathbb{Z}^{m \times n}$be an integer matrix of rank$n$. I want to prove that$\mathrm{det}(B^TB)$is a perfect square. Of course if$m=n$,$\mathrm{det}(B^TB) = \mathrm{det}(B)^2$... • 12.3k 0 votes 0 answers 50 views ### Rotation property of dual lattice The dual lattice is defined like this: $$\Lambda^* = \{v \in \text{span}_\mathbb{R}(\Lambda): \langle v, w \rangle \in \mathbb{Z} \, \text{for all } w \in \Lambda\}$$ The dual lattice has the ... • 204 1 vote 0 answers 29 views ### Two variable Ehrhart polynomials Is it possible to count the number of lattice points in a polytope that is scaled unevenly in two dimensions (or more generally mapped by some linear map) by some kind of two variable Ehrhart ... • 7,659 0 votes 1 answer 33 views ### Blichfeldt vs Minkowski Theorems (Mathematics of Lattices) I am trying to study and understand the basic theorems about mathematics of lattices. In particular, I understood statements and proofs of both Blichfeldt and Minkowski. My doubt was about their ... 0 votes 1 answer 50 views ### Given a basis set of a lattice and the normalized shortest vector of this lattice, can we solve SVP efficiently (i.e. in poly time)? Given a basis set B (say$m \times n$), denote the lattice by L(B), also given a unit vector$\vec{d} = \frac{1}{\lambda_1}\vec{u}$, say$\vec{u}$is the unique shortest vector of L(B). In other words,... 0 votes 0 answers 28 views ### convex polyhedral cone interpretation According to my text, a convex polyhedral cone in$N_{\mathbb{R}}$, is a set of the form$\sigma = \big \lbrace \: a_1n_1+\cdots +a_rn_r \mid a_i\in \mathbb{R}_{\geq 0} \: \big \rbrace$generated by a ... • 325 0 votes 1 answer 25 views ### Are these two conditions always equivalent for the lattice$\mathbb Z^d$for any property (P)? Consider the lattice$\mathbb Z^d$in$\mathbb R^d$and the following two statements: Let$(P)$be a property on vectors in$\mathbb R^d$and let$1\le k \le d$. (1) For any$k$linearly independent ... • 379 0 votes 0 answers 23 views ### generator of the lattice$L\subset \Bbb{C}$Let$L\subset \Bbb{C}$be a lattice, assume there is a$\gamma \in \Bbb{C}$such that$\gamma L = L$, prove the following two facts: (1)$\gamma$must be roots of unity (2) if$\gamma$is not real, ... • 4,284 0 votes 1 answer 39 views ### Standard reference for a fundamental theorem on classification of root lattices In Schuett-Shioda's Mordell-Weil Lattices, the authors refer to a fundamental theorem on root lattices: Theorem 2.25 Any positive-definite even integral root lattice is isometric to an orthogonal sum ... • 9,309 1 vote 0 answers 24 views ### Number of elements in$\text{SL}(d,\mathbb Z)$with bounded$2$-norm I wonder if there are good estimate for the number of elements in$\text{SL}(d,\mathbb Z)$with$2$-norm bounded by$T$as$T \to \infty$, namely $$\#\{g\in SL(d,\mathbb Z):\|g\|_2:=\sqrt{ \sum_{ij}g_{... • 379 0 votes 1 answer 24 views ### Upper Bound on the shortest non-zero non-negative vector in an integer lattice Suppose we have an integer lattice of full rank L\subseteq\Bbb{Z}^n, say of determinant D. Do we have a nice bound on the smallest non-zero vector x\in L s.t. x_i\geq0 for all i? Perhaps ... • 746 0 votes 0 answers 40 views ### Estimating number of lattice points on a hyperbola Given a hyperbola$$y = x + \frac n x \tag 1$$and a range$x \in (a,b)$, how do we find the number of lattice points lying in the range that the hyperbola passes through? A lattice point is a point$(...
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Let $W$ be a linear subspace of $\mathbb{R}^n$. Let $\Gamma$ be the $\mathbb{Z}$ span of some set of linearly independent vectors $v_1, \dots, v_k$ in $\mathbb{R}^n$. Is there any ...