Lattice edits

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.

src/Advanced/Lattices/parts/1 minkowski.tex

@@ -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