# Constructing a set with arbitrary strict Hausdorff dimension

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 $\beta$; if $\beta \in \mathbb{Z}$, then constructing a set of strict Hausdorff dimension $\beta$ is indeed trivial.

Suppose $0 < \beta < 1$. Then, we may apply following theorem.

Theorem 1:
Suppose $S_1, \dots, S_m$ are $m$ separated similarities of a self-similar set $F$ with the common ration $r$ that satisfies $0 < r < 1$. Then $F$ has Hausdorff dimension equal to $\log m / \log (1/r)$.

Theorem 1 suggests that Cantor set has Hausdorff dimension of $\log 2 / \log 3$. The proof of that Cantor set has a strict Hausdorff dimension is left to readers.

Let us define a $\mathcal{C}_\delta$ as a Cantor set which was removed relative $\delta$ from each interval for each stage of construction instead of 1/3. Then, Theorem 1 suggests the Hausdorff dimension of $\mathcal{C}_\delta$ is $\log 2 / \log (\frac{1-\delta}{2})$. Since $0 < \delta < 1$, we may choose appropriate $\delta$ to construct $\mathcal{C}_\delta$ of strict Hausdorff dimension $\beta$.

For $\beta > 1$, we may use the idea of $n$-dimensional Cantor dust. $n$-dimensional Cantor dust is constructed by $n$-cartesian product of Cantor sets. The $n$-dimensional Cantor dust has a strict Hausdorff dimension of $n \log 2 / \log 3$, and the proof is left to readers. (Key Idea: Use Theorem 1)

Therefore, for any $\beta > 1$, we may choose a natural number $n$ satisfying $\beta < n < \beta+1$, and find appropriate $\delta$ to render the strict Hausdorff dimension of $\mathcal{C}_\delta$ to be $\beta / n$.

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