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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