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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Proving Binomial Identities Using Bijections To Lattice Paths

How can I derive a bijection to show that the following equality holds? $2\displaystyle\sum\limits_{j=0}^{n-1} \binom{n-1+j}{j} = \binom{2n}{n}$ In class, we've been deriving bijections using ...
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Good textbooks for lattice and coding theory

I am looking for good textbooks for lattice and coding theory. Lattice and coding theory are very interesting on their own, but I have application of the theory to K3 surfaces & modular forms (and ...
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42 views

Multiplications by unimodular matrices

I feel like this must have an obvious answer, but my knowledge of integer arithmetic is limited. Given an (integer) matrix $A$ of dimension $m \times n$ and an unimodular matrix $U_l$ of dimension $m ...
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Term for intersection of lattice and convex region?

Is there a special term or convenient phrase for the restriction of a convex region to points of a lattice? This is motivated by wanting to talk about the feasible points of a discrete problem. I'd ...
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Is there an algorithm to find a basis for the lattice $V \cap \Bbb{Z}^n$ given a basis for $V \subseteq \Bbb{Q}^n$?

This might be a stupid/very simple question, but since I can't quite seem to come up with a nice trick I will ask it anyway. Assume that we have a vectorspace $V \subseteq \mathbb{Q}^n$ given in the ...
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Finding a basis for the solution space of a system of Diophantine equations

Let $m$, $n$, and $q$ be positive integers, with $m \ge n$. Let $\mathbf{A} \in \mathbb{Z}^{n \times m}_q$ be a matrix. Consider the following set: $S = \big\{ \mathbf{y} \in \mathbb{Z}^m \mid ...
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How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
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Probability distribution of a self avoiding walk

Preliminary: Consider a walk on the lattice $\mathbb{Z}_d$ lattice of length $N$. In a normal random walk, if we let $N$ get large the end position has a probability distribution (PDF) that looks ...
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Conics passing through integer lattice points [closed]

Find parabola that passes through exactly 3 lattice points. Find parabola that passes through exactly 4 lattice points. Prove that for any $n\geq 0$, there is a circle that passes through exactly $n$ ...
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275 views

Integral point inside a quadrilateral

Given four points in $\mathbb{R}^2$, is there an efficient method to determine if the convex hull contains an integral point $(m,n)\in\mathbb{Z}^2$? If it helps, I can assume the convex hull is a ...
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171 views

A book useful to learn lattices (discrete groups)

Does anyone know a good book about lattices (as subgroups of a vector space $V$)?
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Bijection between $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ and lattices in $F^n$

I've come across mention of a bijection between lattices in $F^n$ ($F$ a field, in my case $\mathbb{C}(\!(t)\!)$) and elements of $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$, where $O$ is the ring ...
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Norm of the generators of a fractional ideal.

Let $\mathcal{O}_l=\mathbb{Z}[\frac{1+\sqrt{-l}}{2}]$ with $l$ a prime number congruent to 3 mod 4. Let $\mathfrak{a}$ be a non-principal fractional ideal of $\mathcal{O}_l$. My questions are: Why ...
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344 views

How many amplitude levels does a squared QAM constellation have?

In the above question, I choose terminology from communication engineering, but I will precisely define all the involved quantities below. It is a geometric question and I am not sure if it is ...
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A variant of the “closest vector problem” (CVP) in lattice-based cryptography

Consider a public-key scheme on lattices, such as GGH. The private key is a basis $\mathbf{B} \in \mathbb{Z}^{m \times n}$ of a lattice $\mathcal{L}$ with good properties (such as short nearly ...
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Elementary proof that if $A$ is a matrix map from $\mathbb{Z}^m$ to $\mathbb Z^n$, then the map is surjective iff the gcd of maximal minors is $1$

I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by an $m \times n$ matrix $A$, then the map is surjective iff the gcd ...
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Show that the matrix $A$ with integer entries is injective on the reals to $\mathbb{R}^m$ iff it is injective on the integer lattice.

Show that the $m \times n$ matrix $A$ with integer entries is an injective linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ iff it is injective as a linear map from $\mathbb{Z}^n$ to $\mathbb{Z}^m$. ...
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Proving a complicated inequality involving integers

Let $a,b,c,d$ be integers such that $$\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) \mod 2$$ $$ ad-bc =1$$ ...
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3k views

Integer solutions (lattice points) to arbitrary circles

Wolfram Alpha will provide integer solutions to arbitrary circle equations. I'm trying to understand how it's able to calculate them, but despite a fair bit of digging I haven't found any discussion ...
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Complex elliptic curve for the “conjugate” lattice

Let $\Lambda$ be a lattice in $\mathbb{C}$, and $E=\mathbb{C}/\Lambda$ the corresponding complex elliptic curve. Let $\bar{\Lambda}$ be the "conjugate" lattice, i.e. the one obtained by conjugating ...
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148 views

Number of matrices with weakly increasing rows and columns

I'm curious as to how many matrices there are of size $m \times n$ with elements of the set $\{1, \ldots , k\}$ such that each row and column is weakly increasing? The answer should be expressable as ...
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138 views

Bounding the number of integer solutions of the following inequality

Let $r\geq 1$ be a real number, $-1\leq x\leq 1$ a real number and $y>2$ a real number. We consider this data to be fixed. How can I obtain an upper bound on the number of $(a,b,c,d)\in ...
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Smallest angle among two lines in $n \times n$ grid

Given an $n \times n$ grid, what is the minimum angle between any two distinct lines, each going through some grid point $p$ and at least one other grid point? My guess is the minimum is attained by ...
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55 views

If E is a subset of a lattice closed under addition then is the intersection of E with the opposite of some translate finite?

this seems intuitive to me but I'm struggling to prove it (is it false?). Let $E$ be a subset of a lattice (free abelian group of finite rank) closed under addition, containing the origin and such ...
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How can I find the number of the shortest paths between two points on a 2D lattice grid?

How do you find the number of the shortest distances between two points on a grid where you can only move one unit up, down, left, or right? Is there a formula for this? Eg. The shortest path between ...
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51 views

Lower-Bounding angles in integer Lattices

Given an $n \times n$ integer grid I chose any two grid points $a,b$, draw a line $l$ through $a$ and $b$ and measure the angle between $l$ and a horizontal line. I can do this for any grid point pair ...
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299 views

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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310 views

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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108 views

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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721 views

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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354 views

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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176 views

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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447 views

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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137 views

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 ...