After reading this question, I want to know: if $\mathbb{Z}$ is $\mathbb{R}^0$ (according to Hausdorf Dimension of the integer set), and $\mathbb{R}$ is $\mathbb{R}^1$, there is a natural definition of $\mathbb{R}^r$ with $0\leq r \leq 1$?
As I understand, any fractal over $\mathbb{R}$ with hausdorf dimension r would be a potential answer, but which one is the "natural" one? Does Von Neumann's Continuous Geometry explain it, or is another field of math?
I make this question, because I want to understand what is $\mathbb{R}^r$, for example $\mathbb{R}^{0.5}$, and I do not know what I should study. Should I learn about Hyperspheres, Continuous Geometry, more about fractals?