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\begin{document}

\title{Math 18.100B/C, Spring 2012 - Problem Set \# 8}

%\email{jspeck@math.mit.edu}


\maketitle



\fbox{
\begin{minipage}{1 \textwidth}
\textbf{Professors:} Roman Bezrukavnikov and Jared Speck  \
\\
\begin{center}
{\large \textbf{Due: by 3pm on 4-27-12} (in the Undergraduate Math Office: Room 2-108)}
\end{center}
\end{minipage}
}


\bigskip

\textbf{Instructions:} Write your name and whether you
are registered for 18.100B or 18.100C. Students registered for 18.100C are expected to typeset all of their solutions using LaTex.

Collaboration is permitted but you are advised to first attempt
solving the problems yourself, and you must write up the solutions
independently and mention your collaborators and other sources
in your paper.

As always in this course (unless stated otherwise) an answer must be
accompanied by a proof.

\bigskip

\begin{enumerate}

\item (10 points)
Problem 2 on p 138.

\item (15 points) Problem 3 on p 138.


\item (10 points) Problem 5 on p 138.


\item (15 points) Problem 6 on p 138.


\item (15 points) Problem 8 on p 138.


\item (15 points) Problem 12 on p 140.

\item Check that the following curves $\gamma:[0,1]\to \RE^2$ are not rectifiable
 
 \begin{enumerate}
 \item (10 points)
  $\gamma(0)=(0,0)$ and $\gamma(x)=(x,x\sin (\frac{1}{x}))$ for $x\ne 0$.
 
 \item (10 points) $\gamma$ is 
 a {\em space-filling} curve:
 by this we mean that the image of the map
  $\gamma:[0,1]\to \RE^2$
 is the unit square $S=\{(x,y)\in \RE^2\ |\ 0\leq x\leq 1, 0\leq y\leq 1\}$.
 (Example of such a curve is presented in the next problem).
 
 \end{enumerate}
 

\item (Optional, zero points). This problem provides an example of a space filling curve.

(see e.g. wikipedia article on "Hilbert curve" or "space-filling curve" for pictures
illustrating the construction of space-filling curves)

Let $C\subset [0,1]$ be the Cantor set, $S$ be the unit square as in the previous problem
and $C^2\subset S$ be the subset $C^2=\{(x,y)\ |\ x,y\in C\}$.

\begin{enumerate}
\item Check that the map sending $(a_1,a_2,\dots)$ to $\sum \frac{a_n}{3^n}$ is a bijection
from the set of sequences of 0's and 2's to $C$.

[This is essentially presentation of real numbers as infinite ternary fractions, except that a finite ternary fraction
ending at 1, $.a_1a_2\dots a_{n-1}1$ is replaced by $.a_1a_2\dots a_{n-1}02222\dots$.]

We will identify $C$ with the set of sequences of $0$'s and $2$'s.

\item For a finite sequence $a_1,\dots, a_n$ of $0$'s and $2$'s consider the set of infinite sequences
which start with $a_1,\dots, a_n$. Show that this is an open subset in $C$, and that every open
subset in $C$ is a union of such subsets.

\item For a sequence $s=(a_1,a_2\dots)$ of 0's and 2's set $f_1(s)=(a_1,a_3, a_5\dots)$,
$f_2(s)=(a_2,a_4,\dots)$.
Check that the map $f(s)=(f_1(s),f_2(s))$ is a continuous bijection $C\to C^2$.

\item  Show that the map 
sending a sequence $a_1,a_2,\dots$ of 0's and 1's to $\frac{1}{2}\sum \frac{a_n}{2^n}$ 
gives a continuous onto map $\sigma:C\to [0,1]$.

[This map essentially turns a ternary fraction into a binary one by replacing each 2 by 1.
The map "glues together" the endpoints of each interval removed in the process of construction
of $C$.]

\item Deduce that the map $s\mapsto (\sigma(f_1(s)), \sigma(f_2(s)))$ is a continuous
onto map $C\to S$.

\item Now define $\gamma:[0,1]\to S$ so that $\gamma(s)=(\sigma(f_1(s)), \sigma(f_2(s)) )$
when $s\in C$ and $\gamma(s)|_{[a,b]}$ is linear when $a,b\in C$ and $(a,b)\cap C=\emptyset$.
Verify that this requirement defines a continuous onto map $\gamma: [0,1]\to S$. 



\end{enumerate}

\end{enumerate}
\end{document}