Minor edits

Advanced/Lattices/parts/0 intro.tex

@@ -1,5 +1,5 @@
 \definition{}
-The \textit{integer lattice} $\mathbb{Z}^n \subset \mathbb{R}^n$ is the set of points with integer coordinates. We call each point in the lattice a \textit{lattice point}.
+The \textit{integer lattice} $\mathbb{Z}^n \subset \mathbb{R}^n$ is the set of points with integer coordinates.
 
 \problem{}
 Draw $\mathbb{Z}^2$.
@@ -8,11 +8,12 @@ Draw $\mathbb{Z}^2$.
 
 
 \definition{}
-We say a set of vectors $\{v_1, v_2, ..., v_n\}$ \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 uniquely as
 $$
-	a_1v_1 + a_2v_2 + ... + a_nv_n
+	a_1v_1 + a_2v_2 + ... + a_kv_k
 $$
-for integer coefficients $a_i$.
+for integer coefficients $a_i$. \par
+It is fairly easy to show that $k$ must be at least $n$.
 
 \problem{}
 Which of the following generate $\mathbb{Z}^2$?
@@ -29,8 +30,8 @@ Which of the following generate $\mathbb{Z}^2$?
 \vfill
 
 \problem{}
-Find a set of vectors that generates $\mathbb{Z}^2$. \\
-$\{ (0, 1), (1, 0) \} doesn't count.$
+Find a set of two vectors that generates $\mathbb{Z}^2$. \\
+Don't say $\{ (0, 1), (1, 0) \}$, that's too easy.
 
 \vfill
 

Advanced/Lattices/parts/1 minkowski.tex

@@ -1,9 +1,9 @@
 \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 $x - y \in \mathbb{Z}^n$.
 
-\vfill{4mm}
+\vspace{2mm}
 
 Intuitively, this means that you can translate $X$ to cover two lattice points at the same time.
 
@@ -15,8 +15,8 @@ Draw a region in $\mathbb{R}^2$ with volume greater than 1 that contains no latt
 \vfill
 
 
-\problem{Proof in $\mathbb{Z}^2$}
-The following picture gives the idea for the proof of Blichfeldt's theorem. Explain the picture and complete the proof.
+\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.
 
 \begin{center}
 	\includegraphics[angle=90,width=0.5\linewidth]{proof.png}
@@ -59,7 +59,7 @@ A region $X$ is \textit{convex} if the line segment connecting any two points in
 
 
 \definition{}
-We say a region is \textit{symmetric with respect to the origin} if for all points $x$ in the region, $-x$ is also in $X$.
+We say a region $X$ is \textit{symmetric} if for all points $x \in X$, $-x$ is also in $X$.
 
 \problem{}
 \begin{itemize}
@@ -74,13 +74,15 @@ Every convex set in $\mathbb{R}^n$ that is symmetric with respect to the origin
 
 
 \problem{}
-Draw a few sets that satisfy \ref{mink} in $\mathbb{R}^2$. Which is the simplest region that has the properties listed above?
+Draw a few sets that satisfy \ref{mink} in $\mathbb{R}^2$. \par
+What is the simplest region that has the properties listed above?
 
 \vfill
 
 
 \problem{}
-Let $K$ be a region in $\mathbb{R}^2$ satisfying \ref{mink}. Scale this region by $\frac{1}{2}$, called $K' = \frac{1}{2}K$.
+Let $K$ be a region in $\mathbb{R}^2$ satisfying \ref{mink}. \par
+Let $K'$ be this region scaled by $\frac{1}{2}$.
 
 \begin{itemize}
 	\item How does the volume of $K'$ compare to $K$?

Advanced/Lattices/parts/2 orchard.tex

@@ -1,7 +1,6 @@
 \section{Polya's Orchard Problem}
 
-You are standing in the center of a circular orchard of integer radius R. A tree was planted each integer lattice point, and each has grown to the same radius $r$. If the radius is small enough, you will have a clear line of sight through the orchard in certain directions. If the radius is too large, there is no line of sight through the orchard in any direction. See the figure below:
-
+You are standing in the center of a circular orchard of integer radius $R$. A tree of raduis $r$ has been planted at every integer point in the circle. If $r$ is small, you will have a clear line of sight through the orchard. If $r$ is large, there will be no clear line of sight through in any direction:
 
 \begin{center}
 	\hfill
@@ -85,11 +84,19 @@ You are standing in the center of a circular orchard of integer radius R. A tree
 \end{center}
 
 \problem{}
-Show that if $r < \frac{1}{\sqrt{R^2 + 1}}$, you have at least one directon with a clear line of sight. \\
-\hint{Take a look at the ray through the point $(R, 1)$ and calculate the distance from the closest integer points to the ray.}
+Show that you will have at least one clear line of sight if $r < \frac{1}{\sqrt{R^2 + 1}}$. \par
+\hint{Consider the line segment from $(0, 0)$ to $(R, 1)$. Calculate the distance from the closest integer points to the ray.}
 
 \begin{solution}
-	Consider the ray from the origin through the point $(R, 1)$. Clearly, the two closest lattice points are $(1, 0)$ and $(R - 1, 1)$. They are equally far from the ray so let's calculate the distance from $(1, 0)$ to our ray. Call this distance $\delta$. Consider the triangle with vertices $(0, 0)$, $(1, 0)$, and $(R, 1)$. Then the area of this triangle is $\frac{1}{2}$. On the other hand, the area is also given by $\frac{1}{2} \delta \sqrt{R^2 + 1}$. So, $\delta = \frac{1}{\sqrt{R^2+1}}$. Therefore, if $r < \frac{1}{\sqrt{R^2+1}}$, we will have a clear line of sight given by this ray.
+	Consider the ray from the origin to the point $(R, 1)$.
+
+	The two lattice points closest to this ray are $(1, 0)$ and $(R - 1, 1)$. Say the distance from each of these points to the ray is $\delta$. \par
+
+	Now, consider the triangle with vertices $(0, 0)$, $(1, 0)$, and $(R, 1)$. The area of this triangle is $\frac{1}{2}$.
+
+	The area of this triangle is also equal to $\frac{1}{2} \delta \sqrt{R^2 + 1}$. By algebra, $\delta = \frac{1}{\sqrt{R^2+1}}$. \par
+
+	Therefore, if $r < \frac{1}{\sqrt{R^2+1}}$, we will have a clear line of sight given by this ray.
 \end{solution}