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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Closest point to line in lattice

Given an $n$ x $n$ integer grid, I look at all possible lines through two grid points and I am interested in the minimum distance from any grid point (not on the line) to any line. I my guess is that ...
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continued fractions and convex hulls

If I remember correctly, there is a nice correspondence between continued fractions and convex hulls of lattice points in the plane. If $\theta>0$ is the slope of a line in $\mathbb{R}^2$ passing ...
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topology on n-colorings of a lattice?

Consider the collection of all $n$-colorings of $\mathbb{Z^{d}}$ (i.e. the collection of all ways to color each lattice point one of $n$ colors). What are some non-trivial ways to define a topology on ...
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Integer lattice points on a sphere

Suppose we have a sphere centered at the origin of $\mathbb{R^{n}}$ with radius $r$. Are there known theorems that state the number of integer lattice points that lie on the sphere? It seems like this ...
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Question about Pick's Theorem

Is there a Pick's Theorem for a general lattice in $\mathbb{R}^{2}$?
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Relating lattice bases in $\mathbb R^2$

this is a homework question but I am pretty confused on it--just don't know where to start. We're given a lattice basis $(a, b)$ for a lattice $L$ in $\mathbb{R}^2$, and are supposed to show that ...
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Lattice Paths to $(2n, 2n)$ that do not go through $(n,n)$

I know that from the origin to $(x,y)$ there are $${x+y \choose x} = {x+y \choose y}$$ Lattice paths. The question is finding the number of paths from the origin to $(2n, 2n)$ not passing through ...
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What is the “$\tau$” of this elliptic curve

For any $n\geq 1$, let $E_n $ be the elliptic curve given by the equation $y^2 = x(x-1)(x-\zeta_{15^n})$. Here $\zeta_{m} = \exp(2\pi i /m)$ for any positive integer $m$. There is a unique element ...
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Nonzero element of minimal length in a lattice?

Given two linearly independent vectors $a,b\in\mathbb{R}^2$ we form the lattice $L=\{ma+nb|m,n\in\mathbb{Z}\}$. Now, a proof starts with "choose a nonzero vector in $L$ of smallest length...". Why ...
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Lattice Paths Question

You know, how we can have lattice paths, where we can move either one block north, or one block east, and we have the find all the possible ways of reaching the point (x.y) from (0,0). That is ...
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Lattices in complex vector spaces

if I have a finite dimensional complex vector space $V$ with a lattice $\Gamma$ in $V$, then I consider the complex linear span in $V\oplus \overline{V}$ of the elements $\gamma \oplus ...
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Proving this identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$ using lattice paths

How can I prove the identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$? I have to prove it using lattice paths, it should be related to Catalan numbers The $n$th ...
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Construction of a symplectic basis for a lattice

Let $(T,E)$ be a polarized abelian variety ($T=V/L$, $\dim_\mathbb{C} V=g$, $E:V\times V\to\mathbb{R}$ a nondegenerate real alternating bilinear form, with $E(L\times L)\subseteq\mathbb{Z}$ and ...
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“Center-of-Mass” of lattice polygons (generalization of Pick's theorem)

Call a polygon with integer coordinates (in the Euclidean plane) a 'lattice polygon'. Pick's Theorem allows you to efficiently compute the number of lattice points inside this polygon given just its ...
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Which internal angles can a lattice polygon have?

I am wondering if for a lattice polygon an internal angle can take any value? If no which ones not and why? I guess there will be some restrictions due to the discrete nature of the grid but I am ...
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Lattice Reduction of two matrix

I have two matrices A, B with same number of rows. I want lattice reduction on B. During this reduction, I change rows of A accordingly. That is if i-th row and j-th row in B interchanges, swap i-th ...
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Theta series representation of lattices (Leech lattice, especially)

I'm reading now about theta series and its significance in researching self-dual, even lattices. Now, I found the wikipedia article about the Leech lattice, but I'm having trouble understanding where ...
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Convergence of Sum over Integer Lattice

Does the sum $$\sum_{z \in \mathbb{Z}^3\setminus \{(0,0,0)\}} \left( \frac{1}{|{\bf x} - {\bf z}|^2} - \frac{1}{|{\bf z}|^2} \right)$$ converge pointwise or even uniformly for $\varepsilon < |{\bf ...
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Mathematical Results from Counting Points in Lattices

I'm preparing a talk on lattice point enumeration in polytopes (Ehrhart-Macdonald Theory), and I'd like to have an introduction with a few motivational problems/results which arise from the ...
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Lattice simplification

Update: There is an answer on same question I posted on Stack Overflow. I'm working on data structure for graph cut algorithm. Problem is to make different cuts on shortest paths. I made data ...
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Path (Feynman) Integrals over Graphs

I was thinking about Feynman integrals the other day and in particular about discretizing the paths. Does anyone know the lay of the land about what happens when you do path integrals over, say, a ...
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The $n$-shortest lattice vectors problem in $\mathcal{R}^2$

I am looking for an algorithm to compute the $n$ shortest lattice vectors in $\mathcal{R}^2$. The problem statement is as follows: Given a lattice $L: \{ m \vec{u}+n\vec{v} \} \in \mathcal{R}^2$, a ...
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Proof for Minkowski reduced basis

I've read a few articles explaining the way to use the Minkowski reduced basis in a lattice in order to measure the uniformity of the output of a random number generator. However, I can't prove a ...
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Pick's Theorem on a triangular (or hex) grid

Pick's theorem says that given a square grid (that is, all points in the plane with integer coordinates) and a polygon without holes and non selt-intersecting whose vertices are grid points, its area ...