From 50c4d142c5982e65baf2525a3d3fad76931753e0 Mon Sep 17 00:00:00 2001 From: Mark Date: Thu, 3 Apr 2025 19:01:01 -0700 Subject: [PATCH] Lattice edits --- src/Advanced/Lattices/parts/0 intro.tex | 14 ++++---- src/Advanced/Lattices/parts/1 minkowski.tex | 39 +++++++++++++-------- 2 files changed, 33 insertions(+), 20 deletions(-) diff --git a/src/Advanced/Lattices/parts/0 intro.tex b/src/Advanced/Lattices/parts/0 intro.tex index fc03417..2aef2c6 100644 --- a/src/Advanced/Lattices/parts/0 intro.tex +++ b/src/Advanced/Lattices/parts/0 intro.tex @@ -1,5 +1,6 @@ \definition{} -The \textit{integer lattice} $\mathbb{Z}^n \subset \mathbb{R}^n$ is the set of points with integer coordinates. +The \textit{integer lattice} $\mathbb{Z}^n$ is the set of points with integer coordinates in $n$ dimensions. \par +For example, $\mathbb{Z}^3$ is the set of points $(a, b, c)$ where $a$, $b$, and $c$ are integers. \problem{} Draw $\mathbb{Z}^2$. @@ -8,12 +9,13 @@ Draw $\mathbb{Z}^2$. \definition{} -We say a set of vectors $\{v_1, v_2, ..., v_k\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as +We say a set of vectors $\{v_1, v_2, ..., v_k\}$ \textit{generates} $\mathbb{Z}^n$ +if every lattice point can be written as $$ a_1v_1 + a_2v_2 + ... + a_kv_k $$ for integer coefficients $a_i$. \par -It is fairly easy to show that $k$ must be at least $n$. +\textbf{Bonus:} show that $k$ must be at least $n$. \problem{} Which of the following generate $\mathbb{Z}^2$? @@ -30,8 +32,7 @@ Which of the following generate $\mathbb{Z}^2$? \vfill \problem{} -Find a set of two vectors that generates $\mathbb{Z}^2$. \\ -Don't say $\{ (0, 1), (1, 0) \}$, that's too easy. +Find a set of two vectors other than $\{ (0, 1), (1, 0) \}$ that generates $\mathbb{Z}^2$. \\ \vfill @@ -44,7 +45,8 @@ Find a set of vectors that generates $\mathbb{Z}^n$. \pagebreak \definition{} -A \textit{fundamental region} of a lattice is the parallelepiped spanned by a generating set. The exact shape of this region depends on the generating set we use. +Say we have a generating set of a lattice. \par +The \textit{fundamental region} of this set is the $n$-dimensional parallelogram spanned by its members. \par \problem{} Draw two fundamental regions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same. diff --git a/src/Advanced/Lattices/parts/1 minkowski.tex b/src/Advanced/Lattices/parts/1 minkowski.tex index 3c196e6..334c395 100644 --- a/src/Advanced/Lattices/parts/1 minkowski.tex +++ b/src/Advanced/Lattices/parts/1 minkowski.tex @@ -1,6 +1,6 @@ \section{Minkowski's Theorem} -\theorem{Blichfeldt's Theorem} +\theorem{Blichfeldt's Theorem} Let $X$ be a finite connected region. If the volume of $X$ is greater than $1$, $X$ must contain two distinct points that differ by an element of $\mathbb{Z}^n$. In other words, there exist distinct $x, y \in X$ so that $x - y \in \mathbb{Z}^n$. \vspace{2mm} @@ -9,14 +9,22 @@ Intuitively, this means that you can translate $X$ to cover two lattice points a \problem{} -Draw a region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points. Find two points in that region which differ by an integer vector. +Draw a connected region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points. Find two points in that region which differ by an integer vector. \hint{Area is two-dimensional volume.} \vfill \problem{} -The following picture gives an idea for the proof of Blichfeldt's theorem in $\mathbb{Z}^2$. Explain the picture and complete the proof. +Draw a \textit{disconnected} region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points, \par +and show that no two points in that region differ by an integer vector. +\note{In other words, show that \ref{blich} indeed requires a connected region.} + +\vfill + +\problem{} +The following picture gives an idea for the proof of Blichfeldt's theorem in $\mathbb{Z}^2$. \par +Explain the picture and complete the proof. \begin{center} \includegraphics[angle=90,width=0.5\linewidth]{proof.png} @@ -48,10 +56,8 @@ Let $X$ be a region $\in \mathbb{R}^2$ of volume $k$. How many integral points m A region $X$ is \textit{convex} if the line segment connecting any two points in $X$ lies entirely in $X$. \problem{} -\begin{itemize} - \item Draw a convex region in the plane. - \item Draw a region that is not convex. -\end{itemize} +Draw a convex region in two dimensions. \par +Then, draw a two-dimensional region that is \textit{not} convex. \vfill \pagebreak @@ -59,23 +65,28 @@ A region $X$ is \textit{convex} if the line segment connecting any two points in \definition{} -We say a region $X$ is \textit{symmetric} if for all points $x \in X$, $-x$ is also in $X$. +We say a region $X$ is \textit{symmetric with respect to the origin} if for all points $x \in X$, $-x$ is also in $X$. \par +In the following problems, \say{\textit{symmetric}} means \say{symmetric with respect to the origin.} \problem{} -\begin{itemize} - \item Draw a symmetric region. - \item Draw an asymmetric region. -\end{itemize} +Draw a symmetric region. \par +Then, draw an asymmetric region. + +\vfill + +\problem{} +Show that a convex symmetric set always contains the origin. \vfill \theorem{Minkowski's Theorem} -Every convex set in $\mathbb{R}^n$ that is symmetric with respect to the origin and which has a volume greater than $2^n$ contains an integral point that isn't zero. +Every convex set in $\mathbb{R}^n$ that is symmetric and has a volume \par +greater than $2^n$ contains an integral point that isn't zero. \problem{} Draw a few sets that satisfy \ref{mink} in $\mathbb{R}^2$. \par -What is the simplest region that has the properties listed above? +What is a simple class of regions that has the properties listed above? \vfill