Here’s the link to Lee’s Challenge #1:

https://leeconjecture.wordpress.com/2015/03/16/introduction-to-hausdorff-measures-and-fractals/

Consider a non-negative real number ; if , then constructing a set of strict Hausdorff dimension is indeed trivial.

Suppose . Then, we may apply following theorem.

**Theorem 1:**
Suppose

are

separated similarities of a self-similar set

with the common ration

that satisfies

. Then

has Hausdorff dimension equal to

.

**Theorem 1** suggests that Cantor set has Hausdorff dimension of . The proof of that Cantor set has a strict Hausdorff dimension is left to readers.

Let us define a as a Cantor set which was removed relative from each interval for each stage of construction instead of 1/3. Then, **Theorem 1 **suggests the Hausdorff dimension of is . Since , we may choose appropriate to construct of strict Hausdorff dimension .

For , we may use the idea of -dimensional Cantor dust. -dimensional Cantor dust is constructed by -cartesian product of Cantor sets. The -dimensional Cantor dust has a strict Hausdorff dimension of , and the proof is left to readers. (Key Idea: Use **Theorem 1**)

Therefore, for any , we may choose a natural number satisfying , and find appropriate to render the strict Hausdorff dimension of to be .

Thus, we may construct a set with arbitrary Hausdorff dimension.

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[…] Fields: Measure Theory, Real Analysis Proposed: 02.24.15 Solved: 03.16.15 The general outline is presented in the post, CONSTRUCTING A SET WITH ARBITRARY STRICT HAUSDORFF DIMENSION […]

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