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Week 1 Tutorial

The middle-thirds Cantor set is defined as follows. Put \[ C_1 = [\tfrac{0}{3},\tfrac{1}{3}] \cup [\tfrac{2}{3},\tfrac{3}{3}] \] and define $C_{k+1}$ from $C_k$ by removing the open middle-third of every interval in $C_k$. The intersection \[ \mathcal{C} = \bigcap_{n=1}^\infty C_n \] is the middle-thirds Cantor set.

  1. Write down $C_2$ and $C_3$.
  2. Why is $\mathcal{C}$ a Borel subset of $\R$?
  3. Every $x \in [0,1]$ can be written in the form \[ x = \sum_{n=1}^\infty \dfrac{d(n)}{3^n} \] where each $d(n) \in \{0,1,2\}$. Describe in terms of the $d(n)$ what it means for $x$ to belong to $C_1$, to $C_2$, and to $\mathcal{C}$.
  4. Come up with a bijection between $\{0,1\}^\N$ and $\mathcal{C}$.
  5. Prove that $\Lambda(\mathcal{C}) = 0$.

Put \[ S_1 = [\tfrac{0}{8},\tfrac{3}{8}] \cup [\tfrac{5}{8},\tfrac{8}{8}] \] and define $S_{k+1}$ from $S_k$ by removing from every interval in $S_k$ an interval of length $\tfrac{1}{4^{k+1}}$. The intersection \[ \mathcal{S} = \bigcap_{n=1}^\infty S_n \] is a Smith-Volterra-Cantor set.

  1. Write down $S_2$ and $S_3$.
  2. Is there a bijection between $\mathcal{S}$ and $\{0,1\}^\N$?
  3. What is $\Lambda(\mathcal{S})$?