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Lattice edits (#22)
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Mark
2025-04-04 08:36:33 -07:00
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39
src/Advanced/Lattices/parts/1 minkowski.tex
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@ -1,6 +1,6 @@
\section{Minkowski's Theorem}
\theorem{Blichfeldt's Theorem}
\theorem{Blichfeldt's Theorem}<blich>
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}<mink>
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