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Homework 14, Real Analysis 2

Due June 1

Problem 1

  1. Let $latex f:[0,1]\to\R$ satisfy a Hölder condition of exponent $latex \gamma > 1$. Then f is constant.

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

Problem 2

Let $latex K\subset\R$ be the set

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

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

Problem 3

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

  1. Calculate $latex \dim K$

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

Problem 4

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