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