126 lines
4.6 KiB
TeX
126 lines
4.6 KiB
TeX
\section{Minkowski's Theorem}
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\theorem{Blichfeldt's Theorem}<blich>
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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 $x - y \in \mathbb{Z}^n$.
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\vspace{2mm}
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Intuitively, this means that you can translate $X$ to cover two lattice points at the same time.
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\problem{}
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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.
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\hint{Area is two-dimensional volume.}
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\vfill
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\problem{}
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Draw a \textit{disconnected} region in $\mathbb{R}^2$ with volume greater than 1 that contains no lattice points, \par
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and show that no two points in that region differ by an integer vector.
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\note{In other words, show that \ref{blich} indeed requires a connected region.}
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\vfill
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\problem{}
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The following picture gives an idea for the proof of Blichfeldt's theorem in $\mathbb{Z}^2$. \par
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Explain the picture and complete the proof.
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\begin{center}
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\includegraphics[angle=90,width=0.5\linewidth]{proof.png}
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\end{center}
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\begin{solution}
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The fundamental region of $\mathbb{Z}^2$ tiles the plane. Translate these tiles by lattice vectors to stack them on the fundamental region. Then since the union of the intersections of X with these tiles has area greater 1 and they are stacked on a region of area 1, there must be an overlap by a generalization of the pigeonhole principle (if there were no overlap then the sum of the areas would be less than or equal to 1). Take points $x, y$ in the overlap. Then $x - y$ is a lattice point corresponding to the difference in translates, which were lattice points. Hence, $x - y \in \mathbb{Z}^2$.
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\end{solution}
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\vfill
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\pagebreak
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%\problem{}
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%Does your proof of Blichfeldt's theorem in $\mathbb{Z}^2$ extend to a proof of Blichfeldt's theorem in $\mathbb{Z}^n$?
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%\vfill
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\problem{}
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Let $X$ be a region $\in \mathbb{R}^2$ of volume $k$. How many integral points must $X$ contain after a translation?
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\begin{solution}
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$\lceil k \rceil$
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\end{solution}
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\vfill
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\definition{}
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A region $X$ is \textit{convex} if the line segment connecting any two points in $X$ lies entirely in $X$.
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\problem{}
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Draw a convex region in two dimensions. \par
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Then, draw a two-dimensional region that is \textit{not} convex.
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\vfill
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\pagebreak
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\definition{}
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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
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In the following problems, \say{\textit{symmetric}} means \say{symmetric with respect to the origin.}
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\problem{}
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Draw a symmetric region. \par
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Then, draw an asymmetric region.
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\vfill
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\problem{}
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Show that a convex symmetric set always contains the origin.
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\vfill
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\theorem{Minkowski's Theorem}<mink>
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Every convex set in $\mathbb{R}^n$ that is symmetric and has a volume \par
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greater than $2^n$ contains an integral point that isn't zero.
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\problem{}
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Draw a few sets that satisfy \ref{mink} in $\mathbb{R}^2$. \par
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What is a simple class of regions that has the properties listed above?
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\vfill
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\problem{}
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Let $K$ be a region in $\mathbb{R}^2$ satisfying \ref{mink}. \par
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Let $K'$ be this region scaled by $\frac{1}{2}$.
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\begin{itemize}
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\item How does the volume of $K'$ compare to $K$?
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\item Show that the sum of any two points in $K'$ lies in $K$ \hint{Use convexity.}
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\item Apply Blichfeldt's theorem to $K'$ to prove Minkowski's theorem in $\mathbb{R}^2$.
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\end{itemize}
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\vfill
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\problem{}
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Let $K$ be a region in $\mathbb{R}^n$ satisfying \ref{mink}. Scale this region by $\frac{1}{2}$, called $K' = \frac{1}{2}K$.
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\begin{itemize}
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\item How does the volume of $K'$ compare to $K$?
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\item Show that the sum of any two points in $K'$ lies in $K$
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\item Apply Blichfeldt's theorem to $K'$ to prove Minkowski's theorem.
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\end{itemize}
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\begin{solution}
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\begin{itemize}
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\item The volume of $K'$ is $\frac{1}{2^n}$ the volume of $K$.
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\item Take $x, y \in K'$. It follows that $2x, 2y \in K$. Since $K$ is convex, we have that the midpoint of the line segment between $2x$ and $2y$ is in $K$, and so $\frac{2x + 2y}{2} = x + y \in K$.
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\item Since the volume of $K$ is greater than $2^n$, we have the volume of $K'$ is greater than one.
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Applying Blichfeldt's theorem, we can find two distinct points $x, y \in K'$ such that $x - y \in \mathbb{Z}^n$. Since $K'$ is symmetric with respect to the origin, we have that $-y \in K'$. Therefore, $x + (-y) \in K$ by the previous part. $x \neq y, x - y \neq 0$, so we have found a nontrivial integer point in $K$.
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\end{itemize}
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\end{solution}
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\vfill
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\pagebreak |