# Homework 14, Real Analysis 2

## Due June 1

### Problem 1

1. Let $f:[0,1]\to\R$ satisfy a Hölder condition of exponent $\gamma > 1$. Then f is constant.
2. Is $f:[0,1]\to[0,1]\times[0,1]$ is a surjective Hölder function of exponent $\gamma$, then $\gamma \le 1/2$. (Prove directly, without using Lemma 2.2 from the text.)

### Problem 2

Let $K\subset\R$ be the set

$\displaystyle K = \Big\{ \sum_{k=1}^\infty \frac{a_k}{4^k} \in\R : a_k=0\text{ or }2\Big\}$.

Then $\dim K = 1/2$ and $0 < \mathscr H^{1/2}(K) < \infty$.

### Problem 3

Let $2N+1$ be an odd integer and consider the “middle $1/(2N+1)$th” set K, that is, the result of the Cantor process when removing the middle interval of length $1/(2N+1)$ of the previous interval.

1. Calculate $\dim K$
2. Prove that for any $0 < \alpha < 1$, there exists a totally disconnected perfect set in $\R$ whose dimension is larger then $\alpha$.

### Problem 4

There exists a Cantor-like set that has Lebesgue measure zero and Hausdorff measure 1.

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