There exist fractal with similarity dimension between 0 an 1?

Question

How to prove that there exist a fractal with similarity dimension D = x, where x is between 0 and 1?

Answer 1

The dimension of an Iterated Function System fractal can be finetuned. For example, if $C= rC\cup (rC+1)$ with $r\le \frac12$ then the similarity dimension of $C$ is $\frac{\ln 2}{\ln \frac1r}$ and can thus take any value between $0$ and $1$ by suitable choice of $r$

Answer 2

You can for example take the Cantor set as a template. The dimension of the Cantor set is $\ln2/\ln3$. Here the $\ln2$ is because the whole set is two similar parts, and the $\ln3$ is because the diameter of these parts are $1/3$ of the total.

So using the same recipe we can create a set with dimension $D = \ln2/\ln q$ and select $q$ accordingly ($q = e^{\ln2/\ln D} = 2^{1/D}$).

The construction is the same as for the Cantor set instead that instead of removing $1/3$ in the middle of each line segment you remove $1-2\cdot2^{-1/D}$ of the middle (leaving two pieces of length $2^{-1/D}$).